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Chapter Seven The Correlation Coefficient

Copyright © Houghton Mifflin Company. All rights reserved.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.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 7 - 3 As before, 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 Similarly, 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

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 7 - 4 Now, indicates the the sum of the X scores times the sum of the Y scores and indicates that you are to multiply each X score times its associated Y score and then sum the products. New Statistical Notation

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 7 - 5 Correlation Coefficient A correlation coefficient is the descriptive statistic that, in a single number, summarizes and describes the important characteristics in a relationship It does so by simultaneously examining all pairs of X and Y scores

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

Copyright © Houghton Mifflin Company. All rights reserved.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 When a relationship exists, as the X scores increase, the vertical height of the data points changes, indicating that the Y scores are changing

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 7 - 12 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

Copyright © Houghton Mifflin Company. All rights reserved.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.

Copyright © Houghton Mifflin Company. All rights reserved.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 larger the absolute value of the correlation coefficient, the stronger the relationship The sign of the correlation coefficient indicates the direction of a linear relationship

Copyright © Houghton Mifflin Company. All rights reserved.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

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 7 - 21 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. The formula for the Pearson r is

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 7 - 22 The Spearman rank-order correlation coefficient describes the linear relationship between two variables measured using ranked scores. The formula is where N is the number of pairs of ranks and D is the difference between the two ranks in each pair. Spearman Rank-Order Correlation Coefficient

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 7 - 23 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 reduce the accuracy of the correlation coefficient, producing a coefficient that is smaller than it would be if the range were not restricted.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 7 - 24 X Y 18 26 36 45 51 63 Example 1 For the following data set of interval/ratio scores, calculate the Pearson correlation coefficient.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 7 - 25 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.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 7 - 26 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

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 7 - 28 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.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 7 - 29 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.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter 7 - 30 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.