Chapters 8 and 9: Correlations Between Data Sets

Math 1680

Overview Scatter Plots Associations The Correlation Coefficient Sketching Scatter Plots Changes of Scale Summary

Scatter Plots Often, we are interested in comparing

two related data sets Heights and weights of students SAT scores and freshman GPA Age and fuel efficiency of vehicles

We can draw a scatter plot of the data set Plot paired data points on a Cartesian plane

Scatter Plots Scatter plot for

the heights of 1,078 fathers and their adult sons From HANES

study

Scatter Plots

What does the dashed diagonal line represent?

Find the point representing a 5'3¼" father who has a 5'6½" son

Scatter Plots What does the

vertical dashed column represent?

Consider the families where the father was 72" tall, to the nearest inch How tall was the

tallest son? Shortest?

Scatter Plots Was the

average height of the fathers around 64”, 68” or 72”?

Was the SD of the fathers’ heights around 3", 6" or 9"?

Scatter Plots The points form a

swarm that is more or less football-shaped This indicates

that there is a linear association between the fathers’ heights and the sons’ heights

Scatter Plots Short fathers tend

to have short sons, and tall fathers tend to have tall sons We say there is a

positive association between the heights of fathers and sons

What would it mean for there to be a negative association between the heights?

Scatter Plots Does knowing the father’s height

give a precise prediction of his son’s height?

Does knowing the father’s height let you better predict his son’s height?

Scatter Plots We will generally assume the

scatter plots are football-shaped Association is linear in nature Each data set is approximately

normal

Scatter Plots Key features of scatter plots

Given two data sets X and Y, … The point of averages is the point (x, y)

The average of a data set is denoted by μ (Greek mu, for mean)

The subscript indicates which set is being referenced

It will be in the center of the cloud Due to the normal approximation, the vast

majority (95%) of the cloud should fall within 2 SD’s less than and greater than average for both X and Y

Scatter Plots

Associations When given a value in one data

set, we often want to make a prediction for the other data set We call our given value the

independent variable We call the value we are trying to

predict the dependent variable

Associations If there is indeed a relationship between the

two data sets, we can say various things about their association:

Strong: Knowing X helps you a lot in predicting Y, and vice versa

Weak: Knowing X doesn’t really help you predict Y, and vice versa

Positive: X and Y are directly proportional The higher in one you look, the higher in the other you

should be Negative: X and Y are inversely proportional

The higher in one you look, the lower in the other you should be

Associations Positive associations

Study time/final grade Height/weight SAT score/GPA Clouds in sky/chance

of rain Bowling

practice/bowling score Age of husband/age of

wife

Negative associations Age of car/fuel

efficiency Golfing

practice/golf score Dental

hygiene/cavities formed

Pollution/air quality Speed/mile time

Associations What kind of association is this?

Associations What kind of association is this?

Associations Remember that even a very strong

association does not necessarily imply a causal relationship There may be a confounding

influence at play

The Correlation Coefficient While strong/weak and

positive/negative give a sense of the association, we want a way to quantify the strength and direction of the association The correlation coefficient (r) is the

statistic which accomplishes this

The Correlation Coefficient The correlation coefficient is always

between –1 and 1 A positive r means that there is a positive

association between the sets A negative r means that there is a negative

association between the sets If r is close to 0, then there is only a weak

association between the sets If r is close to 1 or –1, then there is a strong

association between the sets

The Correlation Coefficient The following plots have

and , with 50 points in them

The only difference between them is the correlation coefficient Note how the points fall into a line as

r approaches 1 or –1

3 YX 1 YX SDSD

The Correlation Coefficient To calculate r…

Find the average and SD of each data set

Multiply the data sets pairwise and find the average

The correlation is the average of the product minus the product of the averages, all divided by the product of the SD’s

YX

YXXY

SDSDr

The Correlation Coefficient

X Y

1 5

3 9

4 7

5 1

7 13 91

5

28

27

5

XY

4

7

Y

Y

SD

2

4

X

X

SD

2.31XY

4.0)4)(2(

)7)(4(2.31

r

The Correlation Coefficient Compute r for the following data

X Y

1 2

2 1

3 4

4 3

5 7

6 5

7 6

X Y

1 3

3 7

4 9

5 11

7 15

0.8214

1

The Correlation Coefficient Estimate the correlation

The Correlation Coefficient Estimate the correlation

Sketching Scatter Plots The SD line is the line consisting of

all the points where the standard score in X equals the standard score in Y zX = zY

To sketch the SD line, draw a line bisecting the long axis of the football shape Note that the SD line always goes

through the point of averages

Sketching Scatter Plots Given the five-statistic summary (averages,

SD’s, and correlation) for a pair of data sets, we can sketch the scatter plot

Plot the point of averages in the center Mark two SD’s in both directions, on both axes Plot the point 1 SD above average for both data sets draw a line connecting this point and the point of

averages This is the SD line

Draw an ellipse with the SD line as its long axis Ellipse should go just beyond the 2 SD marks in all

directions The value of r determines how oblong the ellipse is

Sketching Scatter Plots A study of the IQs of husbands and wives

obtained the following results Husbands: average IQ = 100, SD = 15 Wives: average IQ = 100,

SD = 15 r = 0.6

Sketch the scatter plot

Changes of Scale The correlation coefficient is not affected by

changes of scale Moving: adding the same number to all of the

values of one variable Stretching: multiplying the same positive

number to all the values of one variable Would r change if we multiplied by a negative number?

The correlation coefficient is also unaffected by interchanging the two data sets

Changes of Scale

Changes of Scale

Changes of Scale Compute r for each of the

following data sets

X Y

0 8

4 9

6 10

8 12

12 6

X Y

0 2

2 3

3 4

4 6

6 0

r = -0.15

Summary The relationship between two variables, X

and Y, can be graphed in a scatter plot When the scatter plot is tightly clustered

around a line, there is a strong linear association between X and Y

A scatter plot can be characterized by its five-statistic summary Average and SD of the X values Average and SD of the Y values Correlation coefficient

Summary When the correlation coefficient gets

closer to 1 or –1, the points cluster more tightly around a line Positive association has a positive r-value Negative association has a negative r-value

Calculating the correlation coefficient Take the average of the product Subtract the product of the averages Divide the difference by the product of the

SD’s

Summary The correlation coefficient is not

affected by changes of scale or transposing the variables

Correlation does not measure causation!