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

SECTION 15.5 | Alternatives to the Pearson Correlation 511
FIGURE 15.13
The relationship between practice and
performance. Although this relationship is
not linear, there is a consistent positive
relationship: an increase in performance
tends to accompany an increase in practice.
Level of performance (Y )
Amount of practice (X)
are on an interval or a ratio scale. As we have noted, the Pearson correlation measures the
degree of linear relationship between two variables—that is, how well the data points fit
on a straight line. However, a researcher often expects the data to show a consistently onedirectional
relationship but not necessarily a linear relationship. For example, Figure 15.13
shows the typical relationship between practice and performance. For nearly any skill,
increasing amounts of practice tend to be associated with improvements in performance
(the more you practice, the better you get). However, it is not a straight-line relationship.
When you are first learning a new skill, practice produces large improvements in performance.
After you have been performing a skill for several years, however, additional
practice produces only minor changes in performance. Although there is a consistent relationship
between the amount of practice and the quality of performance, it clearly is not
linear. If the Pearson correlation were computed for these data, it would not produce a
correlation of 1.00 because the data do not fit perfectly on a straight line. In a situation like
this, the Spearman correlation can be used to measure the consistency of the relationship,
independent of its form.
The reason that the Spearman correlation measures consistency, rather than form,
comes from a simple observation: When two variables are consistently related, their ranks
are linearly related. For example, a perfectly consistent positive relationship means that
every time the X variable increases, the Y variable also increases. Thus, the smallest value
of X is paired with the smallest value of Y, the second-smallest value of X is paired with
the second smallest value of Y, and so on. Every time the rank for X goes up by 1 point, the
rank for Y also goes up by 1 point. As a result, the ranks fit perfectly on a straight line. This
phenomenon is demonstrated in the following example.
EXAMPLE 15.10
TABLE 15.3
Scores and ranks for
Example 15.10.
Table 15.3 presents X and Y scores for a sample of n = 4 people. Note that the data show a
perfectly consistent relationship. Each increase in X is accompanied by an increase in Y. However
the relationship is not linear, as can be seen in the graph of the data in Figure 15.14(a).
Person X Y X-Rank Y-Rank
A 2 2 1 1
B 3 8 2 2
C 4 9 3 3
D 10 10 4 4