Understanding Correlation Coefficients in Statistical Analysis

1. Chapter Seven The Correlation Coefficient

2. More Statistical Notation Correlational analysis requires scores from two variables. X stands for the scores on one variable and Y stands for the scores on the other variable. Usually, each pair of XY scores is from the same participant. Chapter 7 - 2

3. New Statistical Notation • indicates the sum of the X scores, indicates the sum of the squared X scores, and indicates the square of the sum of the X scores • indicates the sum of the Y scores, indicates the sum of the squared Y scores, and indicates the square of the sum of the Y scores Chapter 7 - 3

4. indicates the sum of the X scores times the sum of the Y scores and indicates you are to multiply each X score times its associated Y score and then sum the products New Statistical Notation Chapter 7 - 4

5. Correlation Coefficient A correlation coefficient is the descriptive statistic that, in a single number, summarizes and describes the important characteristics of a relationship Chapter 7 - 5

6. Understanding Correlational Research Chapter 7 - 6

7. Drawing Conclusions The term correlation is synonymous with relationship However, the fact there is a relationship between two variables does not mean that changes in one variable cause the changes in the other variable Chapter 7 - 7

8. Plotting Correlational Data A scatterplot is a graph that shows the location of each data point formed by a pair of X-Y scores A data point that is relatively far from the majority of data points in a scatterplot is called an outlier Chapter 7 - 8

9. A Scatterplot Showing the Existence of a Relationship Between the Two Variables Chapter 7 - 9

10. Types of Relationships Chapter 7 - 10

11. Linear Relationships In a linear relationship, as the X scores increase, the Y scores tend to change in only one direction In a positive linear relationship, as the scores on the X variable increase, the scores on the Y variable also tend to increase In a negative linear relationship, as the scores on the X variable increase, the scores on the Y variable tend to decrease Chapter 7 - 11

12. Linear Relationships The regression line summarizes a relationship by passing through the center of the scatterplot. Chapter 7 - 12

13. A Scatterplot of a Positive Linear Relationship Chapter 7 - 13

14. A Scatterplot of a Negative Linear Relationship Chapter 7 - 14

15. Nonlinear Relationships In a nonlinear, or curvilinear, relationship, as the X scores change, the Y scores do not tend to only increase or only decrease: At some point, the Y scores change their direction of change. Chapter 7 - 15

16. A Scatterplot of a Nonlinear Relationship Chapter 7 - 16

17. Strength of the Relationship Chapter 7 - 17

18. Strength The strength of a relationship is the extent to which one value of Y is consistently paired with one and only one value of X The absolute value of the correlation coefficient indicates the strength of the relationship The sign of the correlation coefficient indicates the direction of a linear relationship (either positive or negative) Chapter 7 - 18

19. Correlation Coefficients Correlation coefficients may range between -1 and +1. The closer to 1 (-1 or +1) the coefficient is, the stronger the relationship; the closer to 0 the coefficient is, the weaker the relationship. As the variability in the Y scores at each X becomes larger, the relationship becomes weaker. Chapter 7 - 19

20. Computing Correlational Coefficients Chapter 7 - 20

21. Data and Scatterplot Reflectinga Correlation Coefficient of 0 Chapter 7 - 21

22. Pearson Correlation Coefficient The Pearson correlation coefficient describes the linear relationship between two interval variables, two ratio variables, or one interval and one ratio variable. Chapter 7 - 22

23. Pearson Correlation Coefficient The formula for the Pearson r is Chapter 7 - 23

24. The Spearman rank-order correlation coefficient describes the linear relationship between two variables measured using ranked scores. Spearman Rank-Order Correlation Coefficient Chapter 7 - 24

25. Spearman Rank-Order Correlation Coefficient • The formula for the Spearman rs is where N is the number of pairs of ranks and D is the difference between the two ranks in each pair Chapter 7 - 25

26. Restriction of Range Restriction of range arises when the range between the lowest and highest scores on one or both variables is limited. This will produce a coefficient that is smaller than it would be if the range were not restricted. Chapter 7 - 26

27. Example 1 For the following data set of interval/ratio scores, calculate the Pearson correlation coefficient. Chapter 7 - 27

28. Example 1Pearson Correlation Coefficient First, we must determine each X2, Y2, and XY value. Then, we must calculate the sum of X, X2, Y, Y2, and XY. Chapter 7 - 28

29. Example 1Pearson Correlation Coefficient Chapter 7 - 29

30. Example 1Pearson Correlation Coefficient Chapter 7 - 30

31. Example 2 For the following data set of ordinal scores, calculate the Spearman rank-order correlation coefficient. Chapter 7 - 31

32. Example 2Spearman Correlation Coefficient First, we must calculate the difference between the ranks for each pair. Chapter 7 - 32

33. Example 2Spearman Correlation Coefficient Next, each D value is squared. Finally, the sum of the D2 values is computed. Chapter 7 - 33

34. Example 2Spearman Correlation Coefficient Chapter 7 - 34

35. Key Terms • positive linear relationship • regression line • restriction of range • scatterplot • Spearman rank-order correlation coefficient • type of relationship correlation coefficient curvilinear relationship linear relationship negative linear relationship nonlinear relationship outlier Pearson correlation coefficient Chapter 7 - 35