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

512 CHAPTER 15 | Correlation
Scores
Ranks
Y scores
10
9
8
7
6
5
4
3
2
1
A
B
C
D
Y ranks
4
3
2
1
A
B
C
D
0
1 2 3 4 5 6 7 8 9 10
X scores
(a)
1 2 3 4
X ranks
(b)
FIGURE 15.14
Scatter plots showing (a) the scores and (b) the ranks for the data in Example 15.10. Notice that there is a consistent,
positive relationship between the X and Y scores, although it is not a linear relationship. Also notice that the scatter plot
for the ranks shows a perfect linear relationship.
Next, we convert the scores to ranks. The lowest X is assigned a rank of 1, the next
lowest a rank of 2, and so on. The Y scores are then ranked in the same way. The ranks are
listed in Table 15.3 and shown in Figure 15.14(b). Note that the perfect consistency for the
scores produces a perfect linear relationship for the ranks.
■
The word monotonic
describes a sequence
that is consistently
increasing (or decreasing).
Like the word
monotonous, it
means constant and
unchanging.
The preceding example demonstrates that a consistent relationship among scores produces
a linear relationship when the scores are converted to ranks. Thus, if you want to
measure the consistency of a relationship for a set of scores, you can simply convert the
scores to ranks and then use the Pearson correlation formula to measure the linear relationship
for the ranked data. The degree of linear relationship for the ranks provides a measure
of the degree of consistency for the original scores.
To summarize, the Spearman correlation measures the relationship between two variables
when both are measured on ordinal scales (ranks). There are two general situations in
which the Spearman correlation is used.
1. Spearman is used when the original data are ordinal; that is, when the X and Y
values are ranks. In this case, you simply apply the Pearson correlation formula to
the set of ranks.
2. Spearman is used when a researcher wants to measure the consistency of a
relationship between X and Y, independent of the specific form of the relationship.
In this case, the original scores are first converted to ranks; then the Pearson
correlation formula is used with the ranks. Because the Pearson formula measures
the degree to which the ranks fit on a straight line, it also measures the degree of
consistency in the relationship for the original scores. Incidentally, when there is
a consistently one-directional relationship between two variables, the relationship
is said to be monotonic. Thus, the Spearman correlation measures the degree of
monotonic relationship between two variables.