Transcript Statistics
Slide 1
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 2
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 3
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 4
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 5
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 6
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 7
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 8
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 9
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 10
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 11
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 12
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 13
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 14
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 15
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 16
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 17
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 18
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 19
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 20
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 21
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 22
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 23
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 24
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 25
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 26
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 27
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 28
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 29
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 30
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 31
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 32
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 33
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 34
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 35
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 36
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 37
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 38
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 39
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 40
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 41
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 42
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 43
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 44
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 45
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 46
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 47
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 48
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 49
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 50
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 51
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 52
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 53
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 54
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 55
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 56
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 57
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 58
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 59
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 60
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 61
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 62
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 63
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 64
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 65
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 66
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 67
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 68
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 69
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 70
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 71
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 72
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 73
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 74
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 75
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 76
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 77
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 78
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 79
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 80
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 81
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 82
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 83
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 84
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 85
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 86
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 87
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 88
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 89
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 90
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 91
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 92
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 93
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 94
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 95
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 96
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 97
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 98
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 99
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 100
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 101
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 102
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 103
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 104
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 2
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 3
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 4
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 5
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 6
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 7
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 8
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 9
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 10
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 11
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 12
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 13
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 14
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 15
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 16
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 17
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 18
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 19
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 20
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 21
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 22
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 23
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 24
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 25
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 26
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 27
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 28
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 29
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 30
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 31
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 32
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 33
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 34
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 35
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 36
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 37
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 38
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 39
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 40
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 41
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 42
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 43
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 44
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 45
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 46
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 47
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 48
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 49
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 50
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 51
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 52
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 53
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 54
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 55
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 56
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 57
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 58
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 59
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 60
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 61
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 62
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 63
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 64
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 65
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 66
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 67
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 68
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 69
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 70
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 71
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 72
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 73
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 74
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 75
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 76
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 77
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 78
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 79
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 80
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 81
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 82
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 83
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 84
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 85
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 86
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 87
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 88
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 89
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 90
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 91
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 92
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 93
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 94
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 95
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 96
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 97
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 98
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 99
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 100
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 101
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 102
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 103
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.
Slide 104
Lincoln Jiang
Statistical Consultant
Western Michigan University
The Graduate College
Graduate Center for Research and Retention
Definition of Statistics
Statistics is the art of making numerical conjectures
about puzzling questions.
--- Statistics Fourth Edition
by Freedman
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Basic Terms
Variables
Characteristics that can take on any number of different
values
Values
Possible numbers or categories that of a variable can
have
Scores
A particular person’s value on a variable
Types of Data
Qualitative data
--nonnumeric
eg: types of material {straw, sticks, bricks}
Quantitative
-- numeric
Discrete data
--numeric data that have a finite number of
possible values
eg: counting numbers, {1,2,3,4,5}
Continuous data
--numeric data that have a infinite number of
possible values
eg: Real numbers
Types of Scale
Nominal---have no order and thus only gives names or labels to
various categories.
Variables assessed on a nominal scale are called categorical variables
Ordinal---have order, but the interval between measurements is not
meaningful.
Interval---have meaningful intervals between measurements, but
there is no true starting point (zero).
Eg: temperature with the Celsius scale
Ratio---have the highest level of measurement. Ratios between
measurements as well as intervals are meaningful because there is a
starting point (zero).
Eg: length, time, plane angle, energy
EX
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Collecting Data
“Twenty-five percent of Americans doubt that the
Holocaust ever occurred.”
--- a news in 1993
Census
Sample Survey
Why Study Samples?
Often not practical to study an entire population
Instead, researchers attempt to make samples
representative of populations
Random selection
Each member of population has an equal chance of being sampled
Good but difficult
Haphazard selection
Take steps to ensure samples do not differ from the population in
systematic ways
Not as good but much more practical
Sample vs. Population
Sample
Relatively small number of
instances that are studied
in order to make
inferences about a larger
group from which they
were drawn
Population
The larger group from
which a sample is drawn
Sample vs. Population Examples
Population
a. pot of beans
b. larger circle
c. histogram
Sample
a. spoonful
b. smaller circle
c. shaded
scores
Sampling Methods
Simple Random Sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other samplings: Quota sampling, Mechanical sampling and so on
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
After Collecting…….Before
Analyzing….
Frequency Tables
Frequency table
Shows how many times each value was used for a
particular variable
Percentage of scores of each value
Grouped frequency table
Range of scores in each of several equally sized intervals
Steps for Making a
Frequency Table
1. Make a list of each possible
value, from highest to lowest
2. Go one by one through the data,
making a mark for each data
next to its value on the list
3. Make a table showing how many
times each value on your list was
used
4. Figure the percentage of data for
each value
A Frequency Table
Stress rating
Frequency
Percent,%
10
14
9.3
9
15
9.9
8
26
17.2
7
31
20.5
6
13
8.6
5
18
11.9
4
16
10.6
3
12
7.9
2
3
2.0
1
1
0.7
0
2
1.3
A Grouped Frequency Table
Stress rating interval
Frequency
Percent
10-11
14
9
8-9
41
27
6-7
44
29
4-5
34
23
2-3
15
10
0-1
3
2
Frequency Graphs
Histogram
Depicts
information from a
frequency table or a
grouped frequency
table as a bar graph
EX2
Shapes of Frequency Distributions
Unimodal
Having one peak
Bimodal
Having two peaks
Multimodal
Having two or more
peaks
Rectangular
Having no peaks
Symmetrical vs. Skewed Frequency
Distributions
Symmetrical distribution
Approximately equal numbers of observations above
and below the middle
Skewed distribution
One side is more spread out that the other, like a tail
Direction of the skew
Right or left (i.e., positive or negative)
Side with the fewer scores
Side that looks like a tail
Skewed Frequency Distributions
Skewed right (b)
Fewer scores right of the peak
Positively skewed
Can be caused by a floor effect
Skewed left (c)
Fewer scores left of the peak
Negatively skewed
Can be caused by a ceiling effect
Ceiling and Floor Effects
Ceiling effects
Occur when scores can go no
higher than an upper limit and
“pile up” at the top
e.g., scores on an easy exam, as
shown on the right
Causes negative skew
Floor effects
Occur when scores can go no
lower than a lower limit and pile
up at the bottom
e.g., household income
Causes positive skew
Kurtosis
Degree to which tails of the distribution are “heavy” or
“light”
heavy tails = higher Kurtosis(b)
Light tails = lower Kurtosis(c)
Normal distribution= Zero Kurtosis (a)
Measures of Central Tendency
Central tendency = representative or typical value in a
distribution
mean, the median and the mode
can measure central tendency.
Mean
Computed by
Summing all the scores (sigma, )
Dividing by the number of scores (N)
M
N
X
Measures of Central Tendency
Mean
Often the best measure of central tendency
Most frequently reported in research articles
Think of the mean as the “balancing point” of the
distribution
Measures of Central Tendency
Mode
Most common single number in a distribution
If distribution is symmetrical and unimodal, the mode =
the mean
Typical way of describing central tendency of a nominal
variable
Measures of Central Tendency
Median
Middle value in a group of scores
Point at which
half the scores are above
half the scores are below
Unaffected by extremity of individual scores
Unlike the mean
Preferable as a measure of central tendency when a
distribution has some extreme scores
Measures of Central Tendency
Examples of means as
balancing points of various
distributions
Does not have to be a score
exactly at the median
Note that a score’s distance
from the balancing point
matters in addition to the
number of scores above or
below it
Measures of Central Tendency
Examples of means and
modes
Measures of Central Tendency
Steps to computing the median
1. Line up scores from highest to lowest
2. Figure out how many scores to the middle
Add 1 to number of scores
Divide by 2
3. Count up to middle score
If there is 1 middle score, that’s the median
If there are 2 middle scores, median is their average
Ex3
Measures of Variation
Variation = how spread out
data is
Variance
Measure of variation
Average of each score’s
squared deviations
(differences) from the mean
Measures of Variation
Steps to computing the variance
1. Subtract the mean from each data
2. Square each deviation value
xi x
( xi x )
3. Add up the squared deviation scores
4. Divide sum by the number of scores
2
(x
i
x)
2
( xi x )
2
n
Ex4
Measures of Variation
Standard deviation
Another measure of variation, roughly the average
amount that scores differ from the mean
Used more widely than variance
Abbreviated as “SD”
To compute standard deviation
Compute variance
Simply take the square root
SD is square root of variance
Variance is SD squared
SD V ariance
2
Two Branches of
Statistical Methods
Descriptive statistics
Summarize and describe a group of numbers such as the
results of a research study
Inferential statistics
Allow researchers to draw conclusions and inferences
that are based on the numbers from a research study,
but go beyond these numbers
The Normal Curve
Often seen in social and behavioral science research
and in nature generally
Particular characteristics
Bell-shaped
Unimodal
Symmetrical
Average tails
Bean Machine
Z Scores
indicates how many standard deviations an
observation is above or below the mean
If Z>0, indicate the data > mean
If Z<0, indicate the data < mean
Z score of 1.0 is one SD above the mean
Z score of -2.5 is two-and-a-half SDs below the mean
Z score of 0 is at the mean
Z
(X M )
SD
Z Scores
When values in a distribution are converted to Z
scores, the distribution will have
Mean of 0
Standard deviation of 1
Useful
Allows variables to be compared to one another
Provides a generalized standard of comparison
Z Scores
To compute a Z score,
subtract the mean from a
raw score and divide by
the SD
To convert a Z score back
to a raw score, multiply
the Z score by the SD
and then add the mean
Z
(X M )
SD
X ( Z )( SD ) M
Ex5
Confidence Interval
confidence interval (CI)
is a particular kind of
interval estimate of a
population parameter.
How likely the interval is
to contain the parameter is
determined by the
confidence level
Animation
ex6
"95% confidence interval"
Correlation
A statistic for describing the relationship between two
variables
Examples
Price of a bottle of wine and its quality
Hours of studying and grades on a statistics exam
Income and happiness
Caffeine intake and alertness
Graphing Correlations on a Scatter
Diagram
Scatter diagram
Graph that shows the degree and
pattern of the relationship between
two variables
Horizontal axis
Usually the variable that does the
predicting
e.g., price, studying, income, caffeine
intake
Vertical axis
Usually the variable that is predicted
e.g., quality, grades, happiness,
alertness
Graphing Correlations on a Scatter
Diagram
Steps for making a
scatter diagram
1. Draw axes and assign
variables to them
2. Determine the range of
values for each variable
and mark the axes
3. Mark a dot for each
person’s pair of scores
Correlation
Linear correlation
Pattern on a scatter diagram is
a straight line
Example above
Curvilinear correlation
More complex relationship
between variables
Pattern in a scatter diagram is
not a straight line
Example below
Correlation
Positive linear correlation
High scores on one variable
matched by high scores on
another
Line slants up to the right
Negative linear correlation
High scores on one variable
matched by low scores on
another
Line slants down to the right
Correlation
Zero correlation
No line, straight or otherwise,
can be fit to the relationship
between the two variables
Two variables are said to be
“uncorrelated”
Correlation Review
a. Negative linear
correlation
b. Curvilinear correlation
c. Positive linear
correlation
d. No correlation
Correlation Coefficient
Correlation coefficient, r, indicates the
precise degree of linear correlation
between two variables
Computed by taking “cross-products”
of Z scores
Multiply Z score on one variable by Z score
on the other variable
Compute average of the resulting products
Can vary from
-1 (perfect negative correlation)
through 0 (no correlation)
to +1 (perfect positive correlation)
r
Z Z
X
N
Y
Linear Correlation Examples
Correlation and Causality
When two variables are
correlated, three possible
directions of causality
X->Y
X<-Y
X<-Z->Y
Inherent ambiguity in
correlations
Knowing that two variables are
correlated tells you nothing
about their causal relationship
Prediction
Correlations can be used to make predictions about
scores
Predictor
X variable
Variable being predicted from
Criterion
Y variable
Variable being predicted
Sometimes called “regression”
Multiple Correlation and
Multiple Regression
Multiple correlation
Association between criterion variables and two or more
predictor variables
Multiple regression
Making predictions about criterion variables based on
two or more predictor variables
Unlike prediction from one variable, standardized
regression coefficient is not the same as the ordinary
correlation coefficient
Proportion of Variance
Accounted For
Correlation coefficients
Indicate strength of a linear relationships
Cannot be compared directly
e.g., an r of .40 is more than twice as strong as an r of .20
To compare correlation coefficients, square them
An r of .40 yields an r2 of .16; an r of .20 an r2 of .04
Squared correlation indicates the proportion of variance
on the criterion variable accounted for by the predictor
variable
R-square
Most Commonly Used Statistical Techniques
Linear Regression (Predicts the value of one numerical
variable given another variable)
- How much does the maximum legibility distance of
Highway signs decrease when age is increased?
Data on winning bid price for 12
Saturn cars on eBaY in July 2002
• Simple linear regression is a data analysis
technique that tries to find a linear pattern
in the data.
•In linear regression, we use all of the data
to calculate a straight line which may be
used to predict Price based on Miles.
• Since Miles is used to predict Price, Miles
is called an `Explanatory (Independent)
Variable' while Price is called a `Response
(Dependent) Variable'.
•The slope of the line is -.05127, which means that predicted Price tends to drop 5
cents for every additional mile driven, or about $512.70 for every 10,000 miles.
•The intercept (or Y-intercept) of the line is $8136; this should not be interpreted
as the predicted price of a car with 0 mileage because the data provides information
only for Saturn cars between 9,300 miles and 153,260 miles
•We can now use the line to predict the selling price of a car with 60000 miles.
What is the height or Y value of the line at X=60000? The answer is
Most Commonly Used Statistical Techniques
T-test (for the means)
- What is the mean time that college students watch TV
per day?
- What is the mean pulse rate of women?
Hypothesis Testing
Procedure for deciding whether the outcome of a study
supports a particular theory or practical innovation
Core Logic of Hypothesis Testing
Approach can seem curious or even backwards
Researcher considers the probability that the
experimental procedure had no effect and that the
observed result could have occurred by chance alone
If that probability is sufficiently low, researcher will…
Reject the notion that experimental procedure had no effect
Affirm the hypothesis that the procedure did have an effect
The Null Hypothesis and the
Research Hypothesis
Null hypothesis (H0)
Opposite of desired result
Usually that manipulation had no effect
Research hypothesis (H1)
Also called the “alternative hypothesis”
Opposite of the null hypothesis
What the experimenter desired or expected all along—
that the manipulation did have an effect
One-tailed vs. Two-tailed
Hypothesis Tests
Directional prediction
Researcher expects experimental procedure to have an
effect in a particular direction
One-tailed significance tests may be used
Nondirectional prediction
Research expects experimental procedure to have an
effect but does not predict a particular direction
Two-tailed significance test appropriate
Takes into account that the sample could be extreme at
either tail of the comparison distribution
One-tailed vs. Two-tailed
Hypothesis Tests
Two-tailed tests
More conservative than one-tailed tests
Some believe that two-tailed tests should always be
used, even when an experimenter makes a directional
prediction
Significance Level Cutoffs for Oneand Two-Tailed Tests
The .05 significance level
The .01 significance level
Decision Errors
When the right procedure leads to the wrong
conclusion
Type I Error
Reject the null hypothesis when it is true
Conclude that a manipulation had an effect when in fact
it did not
Type II Error
Fail to reject the null when it is false
Conclude that a manipulation did not have an effect
when in fact it did
P-value
is the probability of obtaining a result at least as
extreme as the one that was actually observed,
assuming that the null hypothesis is true.
Frequent misunderstandings
For more details, please refer to Wikipedia.
Decision Errors
Setting a strict significance level (e.g., p < .001)
Decreases the possibility of committing a Type I error
Increases the possibility of committing a Type II error
Setting a lenient significance level (e.g., p < .10)
Increases the possibility of committing a Type I error
Decreases the possibility of committing a Type II error
Test Statistic
value computed from sample information
Basis for rejecting/ not rejecting the null hypothesis
used
Example:
to compute the p-value
T-test
A t-test is most
commonly applied when
the test statistic would
follow a normal
distribution. When the
scaling term is unknown
and is replaced by an
estimate based on the
data, the test statistic
follows a Student's t
distribution.
t-test
One-sample t test
Two-sample t test
Independent two-sample
Dependent two-sample
Equal sample size, equal variance
Unequal sample size, equal variance
The Hypothesis Testing Process
1.
2.
3.
4.
5.
6.
Restate the research question as a research
hypothesis and a null hypothesis about the
populations
Set the level of significance, .
Collect the sample and compute for the test statistic.
Assume Ho is true, compute the p-value.
If p-value < , reject Ho.
State your conclusion.
SUMMARY OF HYPOTHESIS TESTS
Ex7,8
Most Commonly Used Statistical Techniques
Analysis of Variance (testing differences of means for 2
or more groups)
- Is GPA related to where a student likes to sit (front,
middle, back)?
- Which internet search engine is the fastest?
Analysis of Variance
Abbreviated as “ANOVA”
Used to compare the means of more than two groups
Null hypothesis is that all populations being studied
have the same mean
Reject null if at least one population has a mean that
differs from the others
Actually works by analyzing variances
Two Different Ways of Estimating
Population Variance
Estimate population variance from variation within
each group
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between
each group
Is affected by whether or not null hypothesis is true
Two Important Questions
1.
How to estimate population variation from variance
between groups?
2.
How is that estimate affected by whether or not the
null is true?
Estimate population variance from variation
between means of groups
First, variation among means
of samples is related directly
to the amount of variation
within each population from
which samples are taken
The more variation within
each population, the more
variation in means of samples
taken from those populations
Note that populations on the
right produce means that are
more scattered
Estimate population variance from variation
between means of groups
And second, when null is false
there is an additional source of
variation
When null hypothesis is true
(left), variation among means of
samples caused by
Variation within the populations
When null hypothesis is false
(right), variation among means
of samples caused by
Variation within the populations
And also by variation among the
population means
Basic Logic of ANOVA
ANOVA entails a
comparison between two
estimates of population
variance
Ratio of between-groups
estimate to within-groups
estimate called an F ratio
Compare obtained F value to
an F distribution
F
Betw een G roups
W ithin G roups
Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even
when those assumptions are violated
Rejecting the Null Hypothesis
A significant F tells you that at least one of the
means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must
conduct follow-up t tests
Problem: Lots of t tests increases the chances of
finding a significant result just by chance (i.e.,
increases chances beyond p = .05)
ANOVA (continue)
Procedure that allows one to examine two or more
variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way
ANOVA, an ANOVA with two variables is a two-way
ANOVA, and so on
Main Effects vs. Interactions
A main effect refers to the effect of one variable,
averaging across the other(s)
An interaction effect refers to a case in which the effect
of one variable depends on the level of another
variable
Main Effects vs. Interactions
Most Commonly Used Statistical Techniques
Chi-square test of independence (Relationship of 2
categorical variables)
-With whom is it easier to make friends with?
- Does the opinion on legalization of marijuana depend
on one’s religion?
Chi-Square Tests
Hypothesis testing procedure for nominal variables
Focus on number of people/items in each category (e.g., hair
color, political party, gender)
Compare how well an observed distribution fits an
expected distribution
Expected distribution can be based on
A theory
Prior results
Assumption of equal distribution across categories
Chi-Square Test for
Goodness of Fit
Single nominal variable
Degrees of freedom = number of categories minus 1
Chi-Square Statistic
Compares observed frequency distribution to expected
frequency distribution
Compute difference between observed and expected and
square each one
Weight each by its expected frequency
Sum them
2
(O E )
2
E
Ex9
Chi-Square Distribution
Compare obtained chi-square
to a chi-square distribution
Does mismatch between
observed and expected
frequency exceed what would
be expected by chance alone?
Chi-Square Test for
Independence
Two nominal variables
Independence means no
relation between
variables
To determine degrees of
freedom…
df ( N C olumn 1)( N R ow s 1)
Contingency table
Lists number of
observations for each
combination of
categories
To determine expected
frequencies…
R
E
(C )
N
Most Commonly Used Statistical Techniques
Correlation (Relationship of 2 numerical variables)
- Is there a connection between the average verbal SAT
and the percent of graduates who took the SAT in a
state?
Other Statistical Techniques
Factor analysis (reducing independent variables which are highly
correlated)
Cluster analysis (grouping observations with similar
characteristics)
Correspondence Analysis (grouping the levels of 2 or more
categorical variables)
Time Series Analysis
And so on……..
Inference with highest confidence level
Definition of Statistics
Statistics is a mathematical science pertaining to the
collection, analysis, interpretation or explanation, and
presentation of data.
---From Wikipedia
Presentation of Data
FOR CATEGORICAL DATA
---Bar Chart
---Pie Chart
Presentation of Data
FOR NUMERICAL DATA
--- Stem-and-Leaf Plot
--- Histogram
--- Boxplot
Overview of Statistical Techniques
Questions?
or
Comments ?
Upcoming Workshops
10/26/2009 Overview of SPSS
12/02/2009
Overview of SAS
How to lie with statistics
1. The Sample with Built-in Bias.
2. Well-Chosen Average.
3. The Gee-Whiz Graph.
4. Correlation and Causation.