Measures of relationship

Correlations

Key Concepts

Pearson Correlation interpretation limits computation graphing

Factors that affect the Pearson Correlation

Coefficient of Determination (r2) – ‘variance explained’

Correlation vs. Causation

Correlations

A correlation measures a linear relationship between two variables

Correlation: Scatterplots

0

20

40

60

80

100

120

0 2 4 6 8 10 12

Age

Weight

Scatterplots are graphic representations of the relationship between two continuous variables

Correlation: Coefficients

Correlation coefficients are number between -1.00 and +1.00 representing the relationship between two variables

0-1 +1

Stop and think

What types of variables are correlated in education?

Can you provide some examples of both positive and negative relationships?

The Ugly Formula

This formula calculates the correlation between X and Y

It builds on your knowledge of variance; showing how the variation in X & Y along with the covariation between X & Y make up the Pearson correlation coefficient.

rX X Y Y

X X Y Y

( )( )

( ) ( )2 2

…the variance formula for r

Example

Step 1: Layout the Problem

Step 2: Compute the Mean for both variables

Sum of X = 58Number of X = 9Mean of X = 6.44

Sum of Y = 530Number of Y = 9Mean of Y = 58.89

Example

Age WeightX X-Xbar Y Y-Ybar7 .56 70 11.114 -2.44 50 -8.899 2.56 100 41.113 -3.44 25 -33.895 1.44 55 -3.894 -2.44 40 -18.896 -.44 75 16.11

10 3.56 90 31.1110 3.56 25 -33.89

Step 3: Compute the difference of each score from its Mean

Mean of X = 6.44 Mean of Y = 58.89

Note: The sum of (X-Xbar) should equal 0 and the sum of (Y-Ybar) should equal 0. Why?

Age WeightX X-Xbar (X-Xbar)2 Y Y-Ybar (Y-Ybar)2

7 .56 .3136 70 11.11 1234.43214 -2.44 5.9536 50 -8.89 79.03219 2.56 6.5536 100 41.11 1690.03213 -3.44 11.8336 25 -33.89 1148.53215 1.44 2.0736 55 -3.89 15.13214 -2.44 5.9536 40 -18.89 356.83216 -.44 .1936 75 16.11 259.5321

10 3.56 12.6736 90 31.11 967.832110 3.56 12.6736 25 -33.89 1148.5321

Step 4: Compute the square of each mean difference

Step 5: Sum the squares differences from the means

Sum (X-Xbar)2 = 58.22 Sum (Y-Ybar)2 = 5788.89

Age WeightX X-Xbar (X-Xbar)2 Y Y-Ybar (Y-Ybar)2 (X-Xbar)(Y-Ybar)7 .56 .3136 70 11.11 1234.4321 6.22164 -2.44 5.9536 50 -8.89 79.0321 21.69169 2.56 6.5536 100 41.11 1690.0321 105.24163 -3.44 11.8336 25 -33.89 1148.5321 116.58165 1.44 2.0736 55 -3.89 15.1321 5.60164 -2.44 5.9536 40 -18.89 356.8321 46.09166 -.44 .1936 75 16.11 259.5321 -7.0884

10 3.56 12.6736 90 31.11 967.8321 110.751610 3.56 12.6736 25 -33.89 1148.5321 -120.6484

Step 6: Compute the cross-product of the differences (for the numerator)

Step 7: Sum the cross product of the differences

Sum (X-Xbar)(Y-Ybar) = 284.4444

Step 8: Collect the partial values together, and substitute each into the formula. Solve the formula.

Sum (X-Xbar)2= 58.22Sum (Y-Ybar)2= 5788.89Sum (X-Xbar)(Y-Ybar) = 284.44

rX X Y Y

X X Y Y

( )( )

( ) ( )2 2

r 2 8 4 4 4

2 2 8 94 9

.

(5 8. )(5 7 8 8. ).

Last Step: Check the computed r for reasonableness, then interpret the value (sign and magnitude)

The value of r must be between -1 and +1 Computed r = .49, which is between -1 and +1

The sign of r is positive The relationship among the two variables is positive “In general, younger people weigh less than older

people.” “In general, older people weigh more than younger

people.”The magnitude of r is “moderate”

Although age and weight are related, the relationship is not very strong. Some of the variation in age has nothing to do with weight, and some of the variation in weight has nothing to do with age.

Variables with a curvilinear relationship will be underestimated if r is applied.

Size of the group does not affect the size of the correlation coefficient.

Cautions

ES = the correlation coefficient, squared (r2)The proportion of the total variance of one

variable that can be associated with the variance in the other variable.

It is the proportion of shared or common variance between two variables.

Example: calorie intake & weightr = .60

r2 = .36 or 36%

Effect Size

CALWEIGHT

r2 = .36

Correlation does not indicate causation correlation indicates a relationship or association

Correlation & Causality

Practice

Compute the Pearson correlation and r squared value for the following example. Be sure to try to draw a rough sketch of a scatterplot to see if the relationship looks linear.

X 3, 7, 8, 2, 5 Y 5, 8, 10, 3, 9

Interpret your results.

x y X-Xbar Y-Ybar (X-Xbar)2 (Y-Ybar) 2 (X-Xbar)(Y-Ybar) - numerator

3 5

7 8

8 10

2 3

5 9

rX X Y Y

X X Y Y

( )( )

( ) ( )2 2

x y X-Xbar Y-Ybar (X-Xbar)2 (Y-Ybar) 2 (X-Xbar)(Y-Ybar)

3 5

7 8

8 10

2 3

5 9Xbar=5

Ybar=7

rX X Y Y

X X Y Y

( )( )

( ) ( )2 2

x y X-Xbar Y-Ybar (X-Xbar)2 (Y-Ybar) 2 (X-Xbar)(Y-Ybar)

3 5 -2 -2

7 8 2 1

8 10 3 3

2 3 -3 -4

5 9 0 -2Xbar=5

Ybar=7 Check=0 Check=0

rX X Y Y

X X Y Y

( )( )

( ) ( )2 2

x y X-Xbar Y-Ybar (X-Xbar)2 (Y-Ybar) 2 (X-Xbar)(Y-Ybar)

3 5 -2 -2 4 4

7 8 2 1 4 1

8 10 3 3 9 9

2 3 -3 -4 9 16

5 9 0 -2 0 4Xbar=5

Ybar=7 Check=0 Check=0 Sum=26 Sum=34

rX X Y Y

X X Y Y

( )( )

( ) ( )2 2

x y X-Xbar Y-Ybar (X-Xbar)2 (Y-Ybar) 2 (X-Xbar)(Y-Ybar)

3 5 -2 -2 4 4 4

7 8 2 1 4 1 2

8 10 3 3 9 9 9

2 3 -3 -4 9 16 12

5 9 0 -2 0 4 0Xbar=5

Ybar=7 Check=0 Check=0 Sum=26 Sum=34 Sum=27

rX X Y Y

X X Y Y

( )( )

( ) ( )2 2

x y X-Xbar Y-Ybar (X-Xbar)2 (Y-Ybar) 2 (X-Xbar)(Y-Ybar)

3 5 -2 -2 4 4 4

7 8 2 1 4 1 2

8 10 3 3 9 9 9

2 3 -3 -4 9 16 12

5 9 0 -2 0 4 0Xbar=5

Ybar=7 Check=0 Check=0 Sum=26 Sum=34 Sum=27

rX X Y Y

X X Y Y

( )( )

( ) ( )2 2

r = 27 / sqrt((26)(34))r = 27 / 29.7r = .908, r2 = .82

Key Points

Correlation is a measure of relationship, and ranges from -1 to 1. Sign indicates direction, and the coefficient indicates strength of relationship.

r2 represents the shared varianceCorrelations do not imply causality