Relationships Between Quantitative Variables

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Transcript Relationships Between Quantitative Variables

Chapter 2

Turning Data Into Information

2.1 Questionnaire

• What is your gender (1=female, 0=male)?

• How many hours did you sleep last night?

• Randomly pick a letter – S or Q.

• Randomly pick a number between 1 and 10.

• What is your height in inches?

• What is your right handspan in centimeters?

• What is your left handspan in centimeters?

• What’s the fastest you’ve ever driven a car?

2.1 Raw Data

•

Raw data

are for numbers and category labels that have been collected but have not yet been processed in any way.

• When measurements are taken from a subset of a population, they represent

sample data.

• When all individuals in a population are measured, the measurements represent

population data.

•

Descriptive statistics:

summary numbers for either population or a sample.

2.2 Types of Data

• Raw data from

categorical variables

consist of group or category names that don’t necessarily have a logical ordering.

Examples

: eye color, country of residence.

• Categorical variables for which the categories have a logical ordering are called

ordinal

variables. Examples : highest educational degree earned, tee shirt size (S, M, L, XL).

• Raw data from

quantitative variables

consist of numerical values taken on each individual.

Examples

: height, number of siblings.

Asking the Right Questions

One Categorical Variable Question 1a:

How many and what percentage of individuals fall into each category?

Example:

What percentage of college students favor the legalization of marijuana, and what percentage of college students oppose legalization of marijuana?

Question 1b:

Are individuals equally divided across categories, or do the percentages across categories follow some other interesting pattern?

Example:

When individuals are asked to choose a number from 1 to 10, are all numbers equally likely to be chosen?

Asking the Right Questions

Two Categorical Variables Question 2a:

Is there a relationship between the two variables, so that the category into which individuals fall for one variable seems to depend on which category they are in for the other variable?

Example:

In Case Study 1.6, we asked if the risk of having a heart attack was different for the physicians who took aspirin than for those who took a placebo.

Question 2b:

Do some combinations of categories stand out because they provide information that is not found by examining the categories separately?

Example:

The relationship between smoking and lung cancer was detected, in part, because someone noticed that the

combination

of being a nonsmoker and having lung cancer is unusual.

Asking the Right Questions

One Quantitative Variable Question 3a:

What are the interesting summary measures, like the average or the range of values, that help us understand the collection of individuals who were measured?

Example:

What is the average handspan measurement, and how much variability is there in handspan measurements?

Question 3b:

Are there individual data values that provide interesting information because they are unique or stand out in some way?

Example:

What is the oldest recorded age of death for a human? Are there many people who have lived nearly that long, or is the oldest recorded age a unique case?

Asking the Right Questions

One Categorical and One Quantitative Variable Question 4a:

Are the measurements similar across categories?

Example:

Do men and women drive at the same “fastest speeds” on average?

Question 4b:

When the categories have a natural ordering (an ordinal variable), does the measurement variable increase or decrease, on average, in that same order?

Example:

Do high school dropouts, high school graduates, college dropouts, and college graduates have increasingly higher average incomes?

Asking the Right Questions

Two Quantitative Variables Question 5a:

If the measurement on one variable is high (or low), does the other one also tend to be high (or low)?

Example:

Do taller people also tend to have larger handspans?

Question 5b:

Are there individuals whose combination of data values provides interesting information because that combination is unusual?

Example:

An individual who has a very low IQ score but can perform complicated arithmetic operations very quickly may shed light on how the brain works. Neither the IQ nor the arithmetic ability may stand out as uniquely low or high, but it is the combination that is interesting.

Explanatory and Response Variables

Many questions are about the

relationship

between

two variables

. It is useful to identify one variable as the

explanatory variable

and the other variable as the

response variable.

In general, the

value of the explanatory variable

for an individual is thought to

partially explain

the

value of the response variable

for that individual.

2.3 Summarizing One or Two Categorical Variables

Numerical Summaries

• Count how many fall into each category.

• Calculate the percent in each category.

• If two variables, have the categories of the explanatory variable define the rows and compute row percentages.

Example 2.1

Importance of Order

Survey of

n = 190

college students. About half (92) given the question: “Randomly pick a letter ---

.” Note: 66% picked the first choice of

Other half (98) given the question: “Randomly pick a letter ---

.” Note: 54% picked the first choice of

Example 2.2

Lighting the Way to Nearsightedness

Survey of

n = 479

children. Those who slept with nightlight or in fully lit room before age 2 had higher incidence of nearsightedness (myopia) later in childhood .

Note

: Study

does not prove

sleeping with light actually

caused

myopia in more children.

Visual Summaries for Categorical Variables

• •

Pie Charts

: useful for summarizing a single categorical variable if not too many categories.

Bar Graphs

: useful for summarizing one or two categorical variables and particularly useful for making comparisons when there are two categorical variables.

Example 2.3

Humans Are Not Good Randomizers

Survey of

n = 190

college students. “Randomly pick a number between 1 and 10.”

Results

: Most chose 7, very few chose 1 or 10.

Example 2.4

Revisiting Nightlights and Nearsightedness

Survey of

n = 479

children.

Response

: Degree of Myopia

Explanatory

2.4 Finding Information in Quantitative Data

Long list of numbers – needs to be organized to obtain answers to questions of interest.

Five-Number Summaries

• Find

extremes

(high, low), the

median

, and the

quartiles

(medians of lower and upper halves of the values). • Quick overview of the data values.

• Information about the center, spread, and shape of data.

Example 2.5

Right Handspans

About 25% of handspans of females are between 12.5 and 19.0 centimeters, about 25% are between 19 and 20 cm, about 25% are between 20 and 21 cm, and about 25% are between 21 and 23.25 cm.

Interesting Features of Quantitative Variables

• • •

Location

: center or average.

e.g. median

Spread

: variability e.g. difference between two extremes or two quartiles.

Shape

: later…

Outliers and How to Handle Them Outlier:

a data point that is not consistent with the bulk of the data.

• Look for them via graphs.

• Can have big influence on conclusions.

• Can cause complications in some statistical analyses.

• Cannot discard without justification.

Example 2.6

Ages of Death of U.S. First Ladies

Partial Data Listing and five-number summary:

Extremes

Possible Reasons for Outliers and Reasonable Actions

•

Mistake made while taking measurement or entering it into computer.

If verified, should be discarded/corrected.

•

Individual in question belongs to a different group than bulk of individuals measured.

Values may be discarded if summary is desired and reported for the majority group only.

•

Outlier is legitimate data value and represents natural variability for the group and variable(s) measured.

Values may not be discarded — they provide important information about location and spread.

Example 2.7

Tiny Boatsmen

Weights (in pounds) of 18 men on crew team:

Cambridge:

188.5, 183.0, 194.5, 185.0, 214.0, 203.5, 186.0, 178.5, 109.0

Oxford:

186.0, 184.5, 204.0, 184.5, 195.5, 202.5, 174.0, 183.0, 109.5

Note:

last weight in each list is unusually small. They are the

coxswains

while others are

rowers

2.5 Pictures for Quantitative Data

•

Histograms

: similar to bar graphs, used for any number of data values.

•

Stem-and-leaf plots

and

dotplots

: present all individual values, useful for small to moderate sized data sets.

•

Boxplot

box-and-whisker plot

: useful summary for comparing two or more groups.

Interpreting Histograms, Stemplots, and Dotplots

• Values are centered around 20 cm.

• Two possible low outliers.

• Apart from outliers, spans range from about 16 to 23 cm.

Describing Shape

• • • •

Symmetric

bell-shaped Symmetric

not bell-shaped Skewed Right

: values trail off to the right

Skewed Left

Creating a Histogram

Step 1:

Decide how many equally spaced (same width)

intervals

to use for the horizontal axis. Between 6 and 15 intervals is a good number.

Step 2:

Decide to use

frequencies

(count) or

relative frequencies

(proportion) on the vertical axis.

Step 3: Draw

equally spaced intervals on horizontal axis covering entire range of the data. Determine frequency or relative frequency of data values in each interval and draw a

bar

with corresponding height

. Decide rule to use for values that fall on the border between two intervals.

2.6 Numerical Summaries of Quantitative Data

Notation for Raw Data:

n =

number of individuals in a data set

1 ,

2 ,

3 ,…,

x n

represent individual raw data values

Example:

A data set consists of handspan values in centimeters for six females; the values are 21, 19, 20, 20, 22, and 19. Then,

n =

1 = 21,

2 = 19,

3 = 20,

4 = 20,

5 = 22, and

• •

Describing the Location of a Data Set

Mean:

the numerical average

Median:

the middle value (if

odd) or the average of the middle two values (

Determining the Mean and Median



The Mean



x i



x n

The Median

is odd:

= middle of ordered values.

Count (

+ 1)/2 down from top of ordered list.

is even:

= average of middle two ordered values.

Average values that are (

/2) and (

/2) + 1 down from top of ordered list.

Example 2.9

Will “Normal” Rainfall Get Rid of Those Odors?

Data

: Average rainfall (inches) for Davis, California for 47 years

Mean

= 18.69 inches

Median

= 16.72 inches In 1997-98, a company with odor problem blamed it on excessive rain.

That year rainfall was 29.69 inches. More rain occurred in 4 other years.

The Influence of Outliers on the Mean and Median

Larger influence on mean

than median.

High outliers will increase the mean. Low outliers will decrease the mean.

If ages at death are: 70, 72, 74, 76, and 78 then mean = median = 74 years.

If ages at death are:

, 72, 74, 76, and 78 then median = 74 but mean = 67 years.

Describing Spread: Range and Interquartile Range

• • •

Range

= high value – low value

Interquartile Range (IQR)

= upper quartile – lower quartile

Standard Deviation

Example 2.10

Fastest Speeds Ever Driven

Five-Number Summary for 87 males

•

Median

= 110 mph measures the center of the data • Two

extremes

describe spread over 100% of data

Range

= 150 – 55 = 95 mph • Two

quartiles

describe spread over middle 50% of data

Interquartile Range

Notation and Finding the Quartiles

Split the ordered values into the half that is below the median and the half that is above the median.

1 =

lower quartile

= median of data values that are

below

the median

3 =

upper quartile

= median of data values that are

above

Example 2.10

Fastest Speeds (cont)

Ordered Data (in rows of 10 values) for the 87 males: 55 60 80 80 80 80 85 85 85 85 90 90 90 90 90 92 94 95 95 95 95

95 100 100 100 100 100 100 100 100 100 101 102 105 105 105 105 105 105 105 105 109

110

110 110 110 110 110 110 110 110 110 110 110 112 115 115 115 115 115 115 120 120 120

120

120 120 120 120 120 120 124 125 125 125 125 125 125 130 130 140 140 140 140 145 150 • •

Median

= (87+1)/2 = 44 th value in the list = 110 mph

= median of the 43 values below the median = (43+1)/2 = 22 nd value from the start of the list = 95 mph •

Percentiles

The

th percentile

is a number that has

% of the data values at or below it and (100 –

)% of the data values at or above it. • Lower quartile = 25 th percentile • Median = 50 th percentile • Upper quartile = 75 th percentile Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.

Picturing Location and Spread with Boxplots

Boxplots for right handspans of males and females.

• Box covers the middle 50% of the data • Line within box marks the median value • Possible outliers are marked with asterisk • Apart from outliers, lines extending from box reach to min and max values.

How to Draw a Boxplot of a Quantitative Variable Step 1:

Label either a vertical axis or a horizontal axis with numbers from min to max of the data.

Step 2:

Draw box with lower end at

1 and upper end at

Step 3:

Draw a line through the box at the median

Step 4:

Draw a line from

1 end of box to smallest data value that is not further than 1.5  IQR from

1. Draw a line from

3 end of box to largest data value that is not further than 1.5  IQR from

Step 5:

Mark data points further than 1.5  IQR from either edge of the box with an asterisk.

Points represented with asterisks are considered to be outliers.

2.7 Bell-Shaped Distributions of Numbers

• Many measurements follow a

predictable pattern

Most

individuals are

clumped

around the

center

• The greater the distance a value is from the center, the fewer individuals have that value.

Variables that follow such a pattern are said to be “

bell-shaped

”. A special case is called a

normal distribution

normal curve

Example 2.11

Bell-Shaped British Women’s Heights

Data

: representative sample of 199 married British couples.

Below shows a histogram of the

wives’ heights

with a normal curve superimposed. The mean height = 1602 millimeters.

Describing Spread with Standard Deviation

Standard deviation

measures variability by summarizing how far individual data values are from the mean.

Think of the standard deviation as

roughly the average distance values fall from the mean

Describing Spread with Standard Deviation

Both sets have same mean of 100.

Set 1: all values are equal to the mean so there is no variability at all.

Set 2: one value equals the mean and other four values are 10 points away from the mean, so the

average distance away from the mean is about

10.

Calculating the Standard Deviation

Formula for the (

sample

)

standard deviation

  

x i n

  1

 2 The value of

2 is called the (

sample

)

variance.

An equivalent formula, easier to compute, is:

 

x i

2 

 1

n x

Calculating the Standard Deviation Step 1: Step 2:

For each observation, calculate the difference between the data value and the mean.

Step 3:

Square each difference in step 2.

Step 4:

Sum the squared differences in step 3, and then divide this sum by

n –

Step 5:

Take the square root of the value in step 4.

Calculating the Standard Deviation

Consider four pulse rates: 62, 68, 74, 76

Step 1:

 62  68  74  76  4 280  70 4

Steps 2 and 3: Step 4: Step 5:

2  120 4  1  40

 40  6 .

Population Standard Deviation

Data sets usually represent a sample from a larger population. If the data set includes measurements for an

entire population

, the notations for the mean and standard deviation are different, and the formula for the standard deviation is also slightly different. A

population mean

is represented by the symbol m (“mu”), and the

population standard deviation

is    

x i n

Interpreting the Standard Deviation for Bell-Shaped Curves: The Empirical Rule

For any bell-shaped curve, approximately •

68%

of the values fall within

1 standard deviation

of the mean in either direction •

95%

of the values fall within

2 standard deviations

of the mean in either direction •

99.7%

of the values fall within

3 standard deviations

The Empirical Rule, the Standard Deviation, and the Range

• Empirical Rule => the range from the minimum to the maximum data values equals about 4 to 6 standard deviations for data with an approximate bell shape. •

You can get a rough idea of the value of the standard deviation by dividing the range by 6. s



Range

Example 2.11

Women’s Heights (cont)

Mean height for the 199 British women is 1602 mm and standard deviation is 62.4 mm.

•

68%

of the 199 heights would fall in the range 1602  62.4, or 1539.6 to 1664.4 mm •

95%

of the heights would fall in the interval 1602  2(62.4), or 1477.2 to 1726.8 mm •

99.7%

Example 2.11

Women’s Heights (cont)

Summary of the actual results: Note: The minimum height = 1410 mm and the maximum height = 1760 mm, for a range of 1760 – 1410 = 350 mm.

So an estimate of the standard deviation is:



Range

6  350 6  58 .

Standardized z-Scores

Standardized score

 Observed or

z-score

: value Standard  Mean deviation

Example:

Mean resting pulse rate for adult men is 70 beats per minute (bpm), standard deviation is 8 bpm. The standardized score for a resting pulse rate of 80:

 80  8 70  1 .

25 A pulse rate of 80 is 1.25 standard deviations

above

the mean pulse rate for adult men.

The Empirical Rule Restated

For bell-shaped data, • About

68%

of the values have

-scores between –1 and +1. • About

95%

of the values have

-scores between –2 and +2. • About

99.7%

of the values have