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

514 CHAPTER 15 | Correlation
The process of finding ranks for tied scores is demonstrated here. These scores have
been listed in order from smallest to largest.
Scores Rank Position Final Rank
3 1 1.5
3 2 1.5
5 3 3
6 4 5
6 5 5
6 6 5
12 7 7
Mean of 1 and 2
Mean of 4, 5, and 6
Note that this example has seven scores and uses all seven ranks. For X = 12, the
largest score, the appropriate rank is 7. It cannot be given a rank of 6 because that rank
has been used for the tied scores.
Caution: In this formula,
you compute the value
of the fraction and then
subtract from 1. The 1 is
not part of the fraction.
EXAMPLE 15.12
■ Special Formula for the Spearman Correlation
After the original X values and Y values have been ranked, the calculations necessary for
SS and SP can be greatly simplified. First, you should note that the X ranks and the Y ranks
are really just a set of integers: 1, 2, 3, 4, . . . , n. To compute the mean for these integers,
you can locate the midpoint of the series by M = (n + 1)/2. Similarly, the SS for this series
of integers can be computed by
SS 5 nsn2 2 1d
sTry it out.d
12
Also, because the X ranks and the Y ranks are the same values, the SS for X is identical to
the SS for Y.
Because calculations with ranks can be simplified and because the Spearman correlation
uses ranked data, these simplifications can be incorporated into the final calculations
for the Spearman correlation. Instead of using the Pearson formula after ranking the data,
you can put the ranks directly into a simplified formula:
r S
5 1 2
6oD2
nsn 2 2 1d
(15.9)
where D is the difference between the X rank and the Y rank for each individual. This
special formula produces the same result that would be obtained from the Pearson formula.
However, note that this special formula can be used only after the scores have been
converted to ranks and only when there are no ties among the ranks. If there are relatively
few tied ranks, the formula still may be used, but it loses accuracy as the number of ties
increases. The application of this formula is demonstrated in the following example.
To demonstrate the special formula for the Spearman correlation, we use the same data that
were presented in Example 15.11. The ranks for these data are shown again here:
Ranks Difference
X Y D D 2
1 5 4 16
2 3 1 1
3 4 1 1
4 2 –2 4
5 1 –4 16
38 = ∑D 2