Pearson-Product Moment Correlation Coefficient (r)

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Pearson-Product Moment Correlation Coefficient (r)
COVxy 
A measure of the relation between x and y, but is not standardized
To standardize , we divide the covariance by the size of the standard deviations.
Given that the maximum value of the covariance is plus or minus the product
of the variance of x and the variance of y, it follows that the limits on the
correlation coefficient are + or – 1.0
r
COVxy
sx s y
Example:
X
(x  x)
Y
3
4
9
4
4
1
10
1
5
0
12
1
6
1
11
0
7
4
13
4
xx  5
sx  158
.
COVxy  2.25
( y  y)
x y  11
sy  158
.
r
2.25
(158
. )(158
. )
r  0.9
COVxy
r
sx s y
Compute the regression coefficient, but using standardized scores.
X’
Y’
-1.27
-1.27
-.066
-.66
0
.66
.66
0
1.27
1.27
b=
Why?
Adjusted r
(1  r 2 )( N  1)
radj  1 
( N  2)
From our example:
radj  1 
= .75
(1.81)(5  1)
(5  2)
r
2
= that proportion of the variance in y that is shared (accounted for) by x.
Sometimes called the “coefficient of determination.”
Thus, r = .9 and
r2
= .81
Or x accounts for 81% of the variance in y.
2
r
2
R = .4, thus r
R = .2, thus
= .04 or 4%
= .016 or 16%
If on r is g times as large as a second r, then the proportion of the
variance associated with the first r will be g(squared) times as
great as that associated with the second.
r
2
can also be misleading
Factors Affecting r
Range Restrictions
Outliers
Heterogeneous
Subsamples
Whole-Part correlations.
This is were the score for variable x contributes to the score of variable y.
Produces a + bias in r.
Again, Correlation does not imply causality. Variables may be accidentally related,
or both may be related to a third variable, or the may influence each other.
Which is more informative, the slope of the regression line
or the correlation coefficient?

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