1. The Pearson Correlation
The Pearson correlation “r” measures the direction and degree of linear (straight line) relationship between two variables.
The magnitude of the Pearson correlation ranges from 0 (indicating no linear relationship between X and Y) to 1.00 (indicating a perfect straight-line relationship between X and Y).
The correlation can be either positive or negative depending on the direction of the relationship.

2. Figure Examples of different values for linear correlations: (a) a perfect negative correlation, –1.00, (b) no linear trend, 0.00, (c) a strong positive relationship, approximately +.90, and (d) a relatively weak negative correlation, approximately –0.40.

3. The Pearson Correlation
degree to which X and Y vary together
r = divided by
degree to which X and Y vary separately
covariability of X and Y
r = divided by
variability of X and Y separately
The Pearson correlation compares the amount of
Covariability of X and Y to the amount X and Y vary separately
If there is a perfect linear relationship
every change in X is matched by a change in the Y variable
see fig 15.3a which illustrates a perfect negative correlation
When X goes up one unit Y goes down one unit
When X goes up two units Y goes down two units
So X and Y covary perfectly

4. The Pearson Correlation
To compute the Pearson correlation
Calculate Covariability which is the sum of products of deviation scores SP = S (X-Mx)(Y-My)
calculate the variability of X and Y scores separately by computing SS for the scores of each variable SSX and SSY
The Pearson correlation is found by computing
the ratio of SP compared to square root of the (SSX)(SSY)
r = SP/(SSX)(SSY)

5. Excel file for calculating correlation coefficient
Example of a perfect positive correlation calculation
Excel file for calculating correlation coefficient

6. The Pearson Correlation Calculations Example 15.1
Calculation of Pearson correlation
r = SP / √ (ssx)(ssy)
r = 2 / √ (ssx)(ssy)
r = 2/ √ (10)(10)
r = 0.20
Note: SS columns are not in the textbook
Calculating SP from definitional formula
SP = S (X-Mx)(Y-My)
SP = (+3)+(-0.5)+(-1.5)+(+1)
SP = +2 X Y
X-Mx
Y-My
Products
(X-Mx)2
(Y-My)2 1 3
-1.5 -2 +3 4 2 6
-0.5 +1
+1.5 -1 7
+0.5 +2
 M=2.5
 M=5
SP = +2
SSx= 10
 SSy= 10

7. The Pearson Correlation Calculations Example 15
The Pearson Correlation Calculations Example 15.1 Using computational formula Not on the exam X Y X2 Y2 XY 1 3 9 2 6 4 36 12 16 7 49 21 10 20 30
110 52
SP Using Computational formula
SP = SXY – (SXSY / n)
SP = [10(20)] /4 = 2
SS Using Computational formula from page 113
SSx = SX2 – (SX)2 /n
SSx = 30 – (10) 2 / 4 = 10
SSy = SY2 – (SY)2 /n
SSy = 110 – (20)2 /4 = 10
Calculation of Pearson correlation
r = SP / √ (ssx)(ssy)
r = 2/ √ (10)(10)
r = 0.20

8. Calculating Sum of Products (SP) Example 15.3
Using definitional formula table 15.1
r = SP/(SSX)(SSY)
r = 28/ (64)(16)
r = 28/32 =

9. Figure 15.4 Scatter plot of data from Example 15.3
r = SP/(SSX)(SSY)
r = 28/ (64)(16)
r = 28/32 = X Y 2 10 6 4 8
Figure Scatter plot of data from Example 15.3

10. Time For More Fun With SPSS

11. Using and Interpreting The Pearson Correlation
Predictions:
knowing the relationship between SAT and GPA
makes it possible to use SAT to predict GPA
Validity:
comparing two tests of the same construct such as “anxiety”
if they have high correlation their is construct validity
Reliability:
Test – Retest reliability
Theory Verification:
When a theory makes a prediction about the relationship between two variables they can be tested with correlation
Amount of sleep is positively related to GPA

12. Interpreting Correlations
Correlations describe relationships
but do not explain why they exist
can not draw cause and effect conclusions
However causation is not ruled out either
Cigarette smoking is positively correlated with cancer
Correlations are sensitive to the range of scores
Correlations are sensitive to outliers
Correlations are not proportions
size of the r value is not directly related to strength of the relationship
use r2 to interpret strength of the relationship

13. Correlations describe relationships
but do not explain why they exist
can not draw cause and effect conclusions
Figure 15-5 Hypothetical data showing the logical relationship between the number of churches and the number of serious crimes for a sample of U.S. cities.

14. Correlations are sensitive to the range of scores
Problem of Restricted Range Figure 15-6 In this example, the green ellipse, when the full range of X and Y values are used there is a strong, positive correlation. However, the brown circle, when the X values have a restricted range of scores the correlation is near zero.

15. Correlations are sensitive to outliers
Problem of Outliers Figure A demonstration of how one extreme data point (an outlier) can influence the value of a correlation.

16. Correlation and Strength of the Relationship
Coefficient of Determination r2
Using correlation for prediction
Using SAT to predict GPA
Based on degree of the relationship
r value is not a good measure for predictions
r2 measures the proportion of variability in one variable that can be determined by the other variable
Small, Medium, Large see table 9.3
Used as a measure of effect size for t test
Amount of variance in the dependent explained by the independent

17. r =
r2 = 0.00
r =
r2 =
r =
r2 =
Figure 15.8 Three sets of data showing three different degrees of linear relationships.

18. Correlation and the Strength of the Relationship
When there is a less-than-perfect correlation between two variables, extreme scores (high or low) for one variable tend to be paired with the less extreme scores (more toward the mean) on the second variable.
This fact is called regression toward the mean.
FIGURE 15.9 A demonstration of regression toward the mean. The figure shows a scatter plot for data with a less-than-perfect correlation. Notice that the highest scores on variable I (extreme right-hand points) are not the highest scores on variable 2, but are displaced downward toward the mean. Also, the lowest scores on variable 1 (extreme left-hand points) are not the lowest scores on variable 2, but are displaced upward toward the mean.

19. The Pearson Correlation and z-scores
Calculations for Pearson Correlation Coefficient
Definitional Formula
r = SP / √ (ssx)(ssy)
SP = S (X-Mx)(Y-My)
Computational Formula
SP = SXY - SXSY / n
z – score formula (for samples)
r = Szxzy / n-1

20. Partial Correlations
Occasionally a researcher may suspect that the relationship between two variables is being distorted by the influence of a third variable.
A partial correlation measures the relationship between two variables while controlling the influence of a third variable by holding it constant.
With three variables, X “churches’, Y “crimes”, and Z “city size” it is possible to compute three individual Pearson correlations:
rXY measuring the correlation between X and Y (churches-crimes)
rXZ measuring the correlation between X and Z (churches-city size)
rYZ measuring the correlation between Y and Z (crimes-city size)

21. Partial Correlations
These three individual correlations can then be used to compute a partial correlation.
For example, the partial correlation between X and Y, holding Z constant, is determined by the following formula
that you do not need to know for the exam
TABLE 15.2 Hypothetical data showing the relationship between the number of churches, the number of crimes, and the populations for a set of n = 15 cities.

22. Hypothetical Data Showing the Logical Relationship
FIGURE Hypothetical data showing the logical relationship between the number of churches and the number of crimes for three groups of cities: those with small populations (Z = 1), those with medium populations (Z = 2), and those with large populations (Z = 3).
FIGURE Hypothetical data showing the logical relationship between the number of churches and the number of crimes for three groups of cities: those with small populations (Z = 1), those with medium populations (Z = 2), and those with large populations (Z = 3).