The Correlation
Coefficient
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Understanding Correlational Research
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Chapter 7 - 2

Correlation Coefficient
A correlation coefficient is the descriptive statistic that, in a single number, summarizes and describes the important characteristics of a 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 - 3

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
© 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

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

Types of Relationships
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Chapter 7 - 6

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
© 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

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 - 8

A Scatterplot of a Negative 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 - 9

Data and Scatterplot Reflecting
No Relationship
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Chapter 7 - 10

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.
© 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

A Scatterplot of a Nonlinear
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 - 12

Strength of the Relationship
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Chapter 7 - 13

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 - 14

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 - 15

Computing Correlational
Coefficients
© 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

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.
© 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

New Statistical Notation
• (

X )(

Y ) indicates the sum of the X scores times the sum of the Y scores and
• 
XY indicates you are to multiply each X score times its associated Y score and then sum the products
© 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

New Statistical Notation
•
•
 X indicates the sum of the X scores,

X
2
( indicates the sum of the squared X scores, and

X )
2 indicates the square of the sum of the
X scores

Y indicates the sum of the Y scores,

Y
2
( indicates the sum of the squared Y scores, and

Y )
2 indicates the square of the sum of the Y scores
© 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

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.
© 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

Pearson Correlation Coefficient
• The formula for the Pearson r is r

N (

XY )

(

X )(

Y )
[ N (

X
2
)

(

X )
2
] [ N (

Y
2
)

(

Y )
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 - 21

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

Spearman Rank-Order
Correlation Coefficient
• The formula for the Spearman r s is r s

1

N
6 (

D
2
)
( N
2 
1 ) where N is the number of pairs of ranks and D is the difference between the two ranks in each pair
© 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 - 23

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 - 24

Linear Relationships
• The regression line summarizes a relationship by passing through the center of the scatterplot.
© 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 - 25

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.
© 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

Example 1
• For the following data set of interval/ratio scores, calculate the Pearson correlation coefficient.
4
5
6
2
3
X
1
Y
6
5
8
6
1
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 - 27

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 .
r

N (

XY )

(

X )(

Y )
[ N (

X
2
)

(

X )
2
] [ N (

Y
2
)

(

Y )
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 - 28

Example 1
Pearson Correlation Coefficient
4
5
6
2
3
X
1
X 2
1
4
9
16
25
36
Y
6
5
8
6
1
3
Y 2
64
36
36
25
1
9

X = 21

X 2 = 91

Y = 29

Y 2 = 171

XY = 81
XY
8
12
18
20
5
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 - 29

Example 1
Pearson Correlation Coefficient r


N (

XY )

(

X )(

Y )
[ N (

X
2
)

(

X )
2
] [ N (

Y
2
)

(

Y )
2
]
6 ( 81 )

( 21 )( 29 )

486
[ 6 ( 91 )

( 21 )
2
] [ 6 ( 171 )

( 29 )
2
]
[ 105

609
] [ 185 ]
 
123
139 .
374
 
0 .
88
© 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 2
• For the following data set of ordinal scores, calculate the Spearman rank-order correlation coefficient.
4
5
6
2
3
X
1
Y
6
4
5
2
3
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 7 - 31

Example 2
Spearman Correlation Coefficient
• First, we must calculate the difference between the ranks for each pair.
Y D
X
1 5 -4 r s

1

N
6 (

D
2
)
( N
2 
1 )
2
3
2
6
0
-3
4 4 0
5 3 2
6 1 5
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Chapter 7 - 32

Example 2
Spearman Correlation Coefficient
• Next, each
D value is squared.
• Finally, the sum of the D 2 values is computed.
• ∑
D 2 =54
4
5
6
2
3
X
1
Y
6
4
5
2
3
1
D
-4
0
-3
0
2
5
D 2
16
0
9
0
4
25
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Chapter 7 - 33

Example 2
Spearman Correlation Coefficient r s

1

N
6 (

D
2
)
( N
2 
1 )

1

6 (
6 ( 54 )
36

1 )


1

324
1

1 .
210
54
 
0 .
54
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Chapter 7 - 34

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