Chapter Seven
The Correlation
Coefficient
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 - 2
New Statistical Notation
• As before, X indicates the sum of the X
2
scores, X indicates the sum of the
2
squared X scores, and (X ) indicates the
square of the sum of the
X scores
• Similarly, Y indicates the sum of the Y
scores, Y 2 indicates the sum of the
2
squared Y scores, and (Y ) indicates the
square of the sum of the Y scores
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Chapter 7 - 3
New Statistical Notation
Now, (X )(Y ) indicates the the sum of
the X scores times the sum of the Y
scores and XY indicates that you are to
multiply each X score times its associated
Y score and then sum the products.
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Chapter 7 - 4
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
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Chapter 7 - 5
Understanding Correlational Research
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Chapter 7 - 6
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
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Chapter 7 - 7
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
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Chapter 7 - 8
A Scatterplot Showing the Existence of a
Relationship Between the Two Variables
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Chapter 7 - 9
Scatterplots Showing No 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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Chapter 7 - 12
A Scatterplot of a Positive Linear
Relationship
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Chapter 7 - 13
A Scatterplot of a Negative Linear
Relationship
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Chapter 7 - 14
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 - 15
A Scatterplot of a Nonlinear Relationship
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Chapter 7 - 16
Strength of the Relationship
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Chapter 7 - 17
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
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Chapter 7 - 18
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
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Chapter 7 - 19
Computing Correlational
Coefficients
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Chapter 7 - 20
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
N (XY )  (X )(Y )
r
2
2
2
2
[ N (X )  (X ) ][ N (Y )  (Y ) ]
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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. The formula is
6(D )
rs  1 
2
N ( N  2)
2
where N is the number of pairs of ranks and D
is the difference between the two ranks in
each pair.
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Chapter 7 - 22
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.
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Chapter 7 - 23
Example 1
• For the following data
set of interval/ratio
scores, calculate the
Pearson correlation
coefficient.
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X
Y
1
8
2
6
3
6
4
5
5
1
6
3
Chapter 7 - 24
Example 1
Pearson 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.
r
N (XY )  (X )(Y )
[ N (X 2 )  (X ) 2 ][ N (Y 2 )  (Y ) 2 ]
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Chapter 7 - 25
Example 1
Pearson Correlation Coefficient
X
X2
Y
Y2
XY
1
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
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XY = 81
Chapter 7 - 26
Example 1
Pearson Correlation Coefficient
r

N (XY )  (X )( Y )
[ N (X 2 )  (X ) 2 ] [ N (Y 2 )  (Y ) 2 ]
6(81)(21)( 29 )
[6(91)  (21) 2 ] [6(171)  (29 ) 2 ]

486  609
[105 ] [185 ]
123

  0.88
139 .374
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Chapter 7 - 27
Example 2
• For the following data
set of ordinal scores,
calculate the Spearman
rank-order correlation
coefficient.
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X
Y
1
5
2
2
3
6
4
4
5
3
6
1
Chapter 7 - 28
Example 2
Spearman Correlation Coefficient
• First, we must calculate the difference
between the ranks for each pair.
6(D )
rs  1 
2
N ( N  2)
2
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X
Y
D
1
5
-4
2
2
0
3
6
-3
4
4
0
5
3
2
Chapter 7 - 29
Example 2
Spearman Correlation Coefficient
• Next, each D value
is squared.
• Finally, the sum of
the D2 values is
computed.
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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
Chapter 7 - 30
Example 2
Spearman Correlation Coefficient
6(D )
rs  1 
2
N ( N  2)
2
6(29)
1
5(25  2)
174
1
115
 1 1.513  0.51
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Chapter 7 - 31