Transcript Educational Research Chapter 11 Descriptive Statistics Gay, Mills, and Airasian

Educational Research
Chapter 11
Descriptive Statistics
Gay, Mills, and Airasian
Topics Discussed in this Chapter
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Preparing data for analysis
Types of descriptive statistics
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Central tendency
Variation
Relative position
Relationships
Calculating descriptive statistics
Preparing Data for Analysis
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Issues
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Scoring procedures
Tabulation and coding
Use of computers
Scoring Procedures
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Instructions
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Standardized tests detail scoring instructions
Teacher-made tests require the delineation of
scoring criteria and specific procedures
Types of items
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Selected response items - easily and objectively
scored
Open-ended items - difficult to score objectively
with a single number as the result
Objectives 1.1 & 1.2
Tabulation and Coding
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Tabulation is organizing data
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Identifying all information relevant to the analysis
Separating groups and individuals within groups
Listing data in columns
Coding
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Assigning names to variables
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EX1 for pretest scores
SEX for gender
EX2 for posttest scores
Objectives 2.1, 2.2, & 2.3
Tabulation and Coding
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Reliability
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Concerns with scoring by hand and
entering data
Machine scoring
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Advantages
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Reliable scoring, tabulation, and analysis
Disadvantages
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Use of selected response items, answering on
scantrons
Objectives 1.4 & 1.5
Tabulation and Coding
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Coding
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Assigning identification numbers to
subjects
Assigning codes to the values of nonnumerical or categorical variables
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Gender: 1=Female and 2=Male
Subjects: 1=English, 2=Math, 3=Science, etc.
Names: 001=John Adams, 002=Sally Andrews,
003=Susan Bolton, … 256=John Zeringue
Objectives 2.2 & 2.3
Computerized Analysis
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Need to learn how to calculate descriptive
statistics by hand
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Creates a conceptual base for understanding the
nature of each statistic
Exemplifies the relationships among statistical
elements of various procedures
Use of computerized software
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SPSS-Windows
Other software packages
Objective 2.4
Descriptive Statistics
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Purpose – to describe or summarize
data in a parsimonious manner
Four types
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Central tendency
Variability
Relative position
Relationships
Objective 2.4
Descriptive Statistics
Graphing data – a
frequency polygon
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SCORE
5
Vertical axis
represents the
frequency with which
a score occurs
Horizontal axis
represents the scores
themselves
Frequency
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4
3
2
1
Std. Dev = 1.63
Mean = 6.0
N = 16.00
0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
SCORE
Objectives 3.1 & 3.2
Central Tendency
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Purpose – to represent the typical score
attained by subjects
Three common measures
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Mode
Median
Mean
Objective 4.1
Central Tendency
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Mode
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The most frequently occurring score
Appropriate for nominal data
Median
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The score above and below which 50% of all
scores lie (i.e., the mid-point)
Characteristics
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Appropriate for ordinal scales
Doesn’t take into account the value of each and every
score in the data
Objectives 4.2, 4.3, & 4.4
Central Tendency
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Mean
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The arithmetic average of all scores
Characteristics
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Advantageous statistical properties
Affected by outlying scores
Most frequently used measure of central
tendency
Formula
Objectives 4.2, 4.3, & 4.4
Variability
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Purpose – to measure the extent to
which scores are spread apart
Four measures
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Range
Quartile deviation
Variance
Standard deviation
Objective 5.1
Variability
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Range
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The difference between the highest and
lowest score in a data set
Characteristics
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Unstable measure of variability
Rough, quick estimate
Objectives 5.2 & 5.3
Variability
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Quartile deviation
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One-half the difference between the upper
and lower quartiles in a distribution
Characteristic - appropriate when the
median is being used
Objectives 5.2 & 5.3
Variability
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Variance
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The average squared deviation of all scores
around the mean
Characteristics
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Many important statistical properties
Difficult to interpret due to “squared” metric
Formula
Objectives 5.2 & 5.3
Variability
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Standard deviation
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The square root of the variance
Characteristics
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Many important statistical properties
Relationship to properties of the normal curve
Easily interpreted
Formula
Objectives 5.2 & 5.3
The Normal Curve
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A bell shaped curve reflecting the
distribution of many variables of
interest to educators
See Figure 14.2
See the attached slide
Objective 6.1
The Normal Curve
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Characteristics
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Fifty-percent of the scores fall above the mean
and fifty-percent fall below the mean
The mean, median, and mode are the same
values
Most participants score near the mean; the further
a score is from the mean the fewer the number of
participants who attained that score
Specific numbers or percentages of scores fall
between ±1 SD, ±2 SD, etc.
Objectives 6.1, 6.2, & 6.3
The Normal Curve
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Properties
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Proportions under the curve
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±1 SD = 68%
±1.96 SD = 95%
±2.58 SD = 99%
Cumulative proportions and percentiles
Objectives 6.3 & 6.4
Skewed Distributions
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Positive – many low scores and few high
scores
Negative – few low scores and many high
scores
Relationships between the mean, median,
and mode
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Positively skewed – mode is lowest, median is in
the middle, and mean is highest
Negatively skewed – mean is lowest, median is in
the middle, and mode is highest
Objectives 7.1 & 7.2
Measures of Relative Position
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Purpose – indicates where a score is in
relation to all other scores in the
distribution
Characteristics
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Clear estimates of relative positions
Possible to compare students’
performances across two or more different
tests provided the scores are based on the
same group
Objectives 7.1 & 7.2
Measures of Relative Position
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Types
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Percentile ranks – the percentage of scores
that fall at or above a given score
Standard scores – a derived score based
on how far a raw score is from a reference
point in terms of standard deviation units
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z score
T score
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Stanine
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Objectives 9.3 & 9.4
Measures of Relative Position
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z score
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The deviation of a score from the mean in
standard deviation units
The basic standard score from which all other
standard scores are calculated
Characteristics
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Mean = 0
Standard deviation = 1
Positive if the score is above the mean and negative if it
is below the mean
Relationship with the area under the normal curve
Objective 9.5
Measures of Relative Position
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z score (continued)
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Possible to calculate relative standings like
the percent better than a score, the
percent falling between two scores, the
percent falling between the mean and a
score, etc.
Formula
Objective 9.5
Measures of Relative Position
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T score – a transformation of a z score
where T = 10(z) + 50
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Characteristics
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Mean = 50
Standard deviation = 10
No negative scores
Objective 9.6
Measures of Relative Position
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Stanine – a transformation of a z score
where the stanine = 2(z) + 5 rounded
to the nearest whole number
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Characteristics
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Nine groups with 1 the lowest and 9 the
highest
Categorical interpretation
Frequently used in norming tables
Objective 9.7
Measures of Relationship
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Purpose – to provide an indication of the relationship
between two variables
Characteristics of correlation coefficients
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Strength or magnitude – 0 to 1
Direction – positive (+) or negative (-)
Types of correlation coefficients – dependent on the
scales of measurement of the variables
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Spearman rho – ranked data
Pearson r – interval or ratio data
Objectives 8.1, 8.2, & 8.3
Measures of Relationship
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Interpretation – correlation does not
mean causation
Formula for Pearson r
Objective 8.2
Calculating Descriptive Statistics
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Symbols used in statistical analysis
General rules for calculating by hand
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Make the columns required by the formula
Label the sum of each column
Write the formula
Write the arithmetic equivalent of the
problem
Solve the arithmetic problem
Objectives 10.1, 10.2, 10.3, & 10.4
Calculating Descriptive Statistics
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Using SPSS Windows
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Means, standard deviations, and standard
scores
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The DESCRIPTIVE procedures
Interpreting output
Correlations
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The CORRELATION procedure
Interpreting output
Objectives 10.1, 10.2, 10.3, & 10.4
Calculating Descriptive Statistics
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See the Statistical Analysis of Data module
on the web site for problems related to
descriptive statistics
Formula for the Mean
x

X 
n
Formula for Variance
 x


2
S 
2
x
x
2
N 1
N
Formula for Standard Deviation
 x


2
SD 
x
2
N 1
N
Formula for Pearson Correlation
r
 x  y 
xy 

 x 

 x2 


2
N
N
2




y
 y 2  

 
N 


Formula for z Score
Z
(x  X )
s
x