Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

SECTION 15.2 | The Pearson Correlation 491
You may have noted that the formulas for SP are similar to the formulas you have learned
for SS (sum of squares). The relationship between the two sets of formulas is described in
Box 15.1. The following example demonstrates the calculation of SP with both formulas.
BOX 15.1 Comparing the SP and SS Formulas
It may help you to learn the formulas for SP if you
note the similarity between the two SP formulas and
the corresponding formulas for SS that were presented
in Chapter 4. The definitional formula for SS is
SS = ∑(X – M) 2
In this formula, you must square each deviation,
which is equivalent to multiplying it by itself.
With this in mind, the formula can be rewritten as
SS = ∑(X – M)(X – M)
The similarity between the SS formula and the SP
formula should be obvious—the SS formula uses
squares and the SP formula uses products. This same
relationship exists for the computational formulas.
For SS, the computational formula is
SS 5 oX 2 2 soXd2
n
As before, each squared value is equivalent to multiplying
by itself, so the formula can be rewritten as
SS 5 oXX 2 oXoX
n
Again, note the similarity in structure between the SS
formula and the SP formula: If you remember that the
SS formulas use squared values and SP formulas use
products, then the two new formulas for the sum of
products should be easy to learn.
EXAMPLE 15.1
Caution: The signs
(1 and 2) are critical
in determining the sum
of products, SP.
The same set of n = 4 pairs of scores are used to calculate SP, first using the definitional
formula and then using the computational formula.
For the definitional formula, you need deviation scores for each of the X values and each
of the Y values. Note that the mean for the Xs is M X
= 2.5 and the mean for the Ys is M Y
= 5.
The deviations and the products of deviations are shown in the following table:
Scores Deviations Products
X Y X – M X
Y – M Y
(X – M X
)(Y – M Y
)
1 3 –1.5 –2 +3
2 6 –0.5 +1 –0.5
4 4 +1.5 –1 –1.5
3 7 +0.5 +2 +1
+2 = SP
For these scores, the sum of the products of the deviations is SP = +2.
For the computational formula, you need the X value, the Y value, and the XY product
for each individual. Then you find the sum of the Xs, the sum of the Ys, and the sum of the
XY products. These values are as follows:
X Y XY
1 3 3
2 6 12
4 4 16
3 7 21
10 20 52 Totals