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 515
Using the special formula for the Spearman correlation, we obtain
r S
5 1 2
6oD2
nsn 2 2 1d
5 1 2 6s38d
5s25 2 1d
5 1 2 228
120
= 1 – 1.90
= – 0.90
This is exactly the same answer that we obtained in Example 15.11, using the Pearson
formula on the ranks.
■
The following example is an opportunity to test your understanding of the Spearman
correlation.
EXAMPLE 15.13
Compute the Spearman correlation for the following set of scores:
X
Y
2 7
12 38
9 6
10 19
You should obtain r S
= 0.80.
■ Testing the Significance of the Spearman Correlation
Testing a hypothesis for the Spearman correlation is similar to the procedure used for the
Pearson r. The basic question is whether a correlation exists in the population. The sample
correlation could be due to chance, or perhaps it reflects an actual relationship between the
variables in the population. For the Pearson correlation, the Greek letter rho (ρ) was used
for the population correlation. For the Spearman, ρ S
is used for the population parameter.
Note that this symbol is consistent with the sample statistic, r S
. The null hypothesis states
that there is no correlation (no monotonic relationship) between the variables for the population,
or in symbols:
H 0
: ρ S
= 0
(The population correlation is zero.)
The alternative hypothesis predicts that a nonzero correlation exists in the population,
which can be stated in symbols as
H 1
: ρ S
≠ 0
(There is a real correlation.)
To determine whether the Spearman correlation is statistically significant (that is, H 0
should be rejected), consult Table B.7. This table is similar to the one used to determine the
significance of Pearson’s r (Table B.6); however, the first column is sample size (n) rather
than degrees of freedom. To use the table, line up the sample size in the first column with
the alpha level at the top. The values in the body of the table identify the magnitude of the
Spearman correlation that is necessary to be significant. The table is built on the concept