1. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter Seven The Correlation Coefficient

2. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 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.

3. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 3 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 New Statistical Notation

4. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 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

5. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 5 Correlation Coefficient A correlation coefficient is the descriptive statistic that, in a single number, summarizes and describes the important characteristics of a relationship

6. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 6 Understanding Correlational Research

7. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 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

8. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 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

9. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 9 A Scatterplot Showing the Existence of a Relationship Between the Two Variables

10. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 10 Types of Relationships

11. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 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

12. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Linear Relationships The regression line summarizes a relationship by passing through the center of the scatterplot. Chapter 7 - 12

13. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 13 A Scatterplot of a Positive Linear Relationship

14. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 14 A Scatterplot of a Negative Linear Relationship

15. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 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.

16. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 16 A Scatterplot of a Nonlinear Relationship

17. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 17 Strength of the Relationship

18. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 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)

19. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 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.

20. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 20 Computing Correlational Coefficients

21. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Data and Scatterplot Reflecting a Correlation Coefficient of 0 Chapter 7 - 21

22. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 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.

23. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Pearson Correlation Coefficient The formula for the Pearson r is Chapter 7 - 23

24. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 24 The Spearman rank-order correlation coefficient describes the linear relationship between two variables measured using ranked scores. Spearman Rank-Order Correlation Coefficient

25. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Spearman Rank-Order Correlation Coefficient The formula for the Spearman r s 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. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 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.

27. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 27 X Y 18 26 36 45 51 63 Example 1 For the following data set of interval/ratio scores, calculate the Pearson correlation coefficient.

28. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 28 Example 1 Pearson Correlation Coefficient First, we must determine each X 2, Y 2, and XY value. Then, we must calculate the sum of X, X 2, Y, Y 2, and XY.

29. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 29 XX2X2 YY2Y2 XY 118648 2463612 3963618 41652520 525115 6363918  X = 21  X 2 = 91  Y = 29  Y 2 = 171  XY = 81 Example 1 Pearson Correlation Coefficient

30. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 30 Example 1 Pearson Correlation Coefficient

31. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 31 X Y 15 22 36 44 53 61 Example 2 For the following data set of ordinal scores, calculate the Spearman rank-order correlation coefficient.

32. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 32 X YD 15-4 220 36-3 440 532 Example 2 Spearman Correlation Coefficient First, we must calculate the difference between the ranks for each pair.

33. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 33 X YDD2D2 15-416 2200 36-39 4400 5324  D 2 = 29 Example 2 Spearman Correlation Coefficient Next, each D value is squared. Finally, the sum of the D 2 values is computed.

34. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 7 - 34 Example 2 Spearman Correlation Coefficient

35. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Key Terms correlation coefficient curvilinear relationship linear relationship negative linear relationship nonlinear relationship outlier Pearson correlation coefficient Chapter 7 - 35 positive linear relationship regression line restriction of range scatterplot Spearman rank-order correlation coefficient type of relationship