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Chapter Seven

The Correlation Coefficient

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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.

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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

X 2X2)( X

Y 2Y2)( Y

New Statistical Notation

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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

))(( YX

XY

New Statistical Notation

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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

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Chapter 7 - 6

Understanding Correlational Research

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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

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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

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Chapter 7 - 9

A Scatterplot Showing the Existence of a Relationship Between the Two Variables

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Chapter 7 - 10

Types of Relationships

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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

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Linear Relationships

• The regression line summarizes a relationship by passing through the center of the scatterplot.

Chapter 7 - 12

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Chapter 7 - 13

A Scatterplot of a Positive Linear Relationship

© 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

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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.

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Chapter 7 - 16

A Scatterplot of a Nonlinear Relationship

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Chapter 7 - 17

Strength of the Relationship

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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)

© 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.

© 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

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Data and Scatterplot Reflectinga Correlation Coefficient of 0

Chapter 7 - 21

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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.

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Pearson Correlation Coefficient

• The formula for the Pearson r is

Chapter 7 - 23

])()([])()([

))(()(2222 YYNXXN

YXXYNr

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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

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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

)2()(61 2

2

NNDrs

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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.

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Chapter 7 - 27

X Y1 8

2 6

3 6

4 5

5 1

6 3

Example 1

• For the following data set of interval/ratio scores, calculate the Pearson correlation coefficient.

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Chapter 7 - 28

])()([])()([

))(()(2222 YYNXXN

YXXYNr

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.

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Chapter 7 - 29

X X2 Y Y2 XY1 1 8 64 8

2 4 6 36 12

3 9 6 36 18

4 16 5 25 20

5 25 1 1 5

6 36 3 9 18

X = 21 X 2 = 91 Y = 29 Y 2 = 171 XY = 81

Example 1Pearson Correlation Coefficient

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Chapter 7 - 30

88.0374.139

123

]185[]105[609486

])29()171(6[])21()91(6[

)29)(21()81(6])()([])()([

))(()(

22

2222

YYNXXN

YXXYNr

Example 1Pearson Correlation Coefficient

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Chapter 7 - 31

X Y1 5

2 2

3 6

4 4

5 3

6 1

Example 2

• For the following data set of ordinal scores, calculate the Spearman

rank-order correlation coefficient.

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Chapter 7 - 32

)2()(61 2

2

NNDrs

X Y D

1 5 -4

2 2 0

3 6 -3

4 4 0

5 3 2

Example 2Spearman Correlation Coefficient

• First, we must calculate the difference between the ranks for each pair.

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Chapter 7 - 33

X Y D D2

1 5 -4 16

2 2 0 0

3 6 -3 9

4 4 0 0

5 3 2 4

D2 = 29

Example 2Spearman Correlation Coefficient

• Next, each D value is squared.

• Finally, the sum of the D2 values is computed.

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Chapter 7 - 34

51.0513.111151741

)225(5)29(61

)2()(61 2

2

NNDrs

Example 2Spearman Correlation Coefficient

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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