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

SECTION 15.2 | The Pearson Correlation 493
Y
6
4
2
FIGURE 15.4
Scatter plot for the data from
Example 15.3.
0 1 2 3 4 5 6 7 8 9 10 X
TABLE 15.1
Calculation of SS X
, SS Y
,
and SP for a sample of
n = 5 pairs of scores.
Scores Deviations Squared Deviations Products
X Y X – M X
Y – M Y
(X – M X
) 2 (Y – M Y
) 2 (X – M X
)(Y – M Y
)
0 2 –6 –2 36 4 +12
10 6 +4 +2 16 4 +8
4 2 –2 –2 4 4 +4
8 4 +2 0 4 0 0
8 6 +2 +2 4 4 +4
SS X
= 64 SS Y
= 16 SP = +28
Using the values from Table 15.1, the Pearson correlation is
SP
r 5
ÏsSS X
dsSS Y
d 5 28
Ïs64ds16d 5 28
32 510.875 ■
■ Correlation and the Pattern of Data Points
Note that the value we obtained for the correlation in Example 15.3 is perfectly consistent
with the pattern formed by the data points in Figure 15.4. The positive sign for the correlation
indicates that the points are clustered around a line that slopes up to the right. Second,
the high value for the correlation (near 1.00) indicates that the points are very tightly
clustered close to the line. Thus, the value of the correlation describes the relationship that
exists in the data.
Because the Pearson correlation describes the pattern formed by the data points, any
factor that does not change the pattern also does not change the correlation. For example,
if 5 points were added to each of the X values in Figure 15.4, then each data point would
move to the right. However, because all of the data points shift to the right, the overall
pattern is not changed, it is simply moved to a new location. Similarly, if 5 points were
subtracted from each X value, the pattern would shift to the left. In either case, the overall
pattern stays the same and the correlation is not changed. In the same way, adding a constant
to (or subtracting a constant from) each Y value simply shifts the pattern up (or down)
but does not change the pattern and, therefore, does not change the correlation. Similarly,
multiplying each X and/or Y value by a positive constant also does not change the pattern
formed by the data points and does not change the correlation. For example, if each of the X
values in Figure 15.4 were multiplied by 2, then same scatter plot could be used to display