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

694 APPENDIX E | Hypothesis Tests for Ordinal Data: Mann-Whitney, Wilcoxon, Kruskal-Wallis, and Friedman Tests
zero-difference scores in the data, it is also recommended that the report describe how they
were treated. For the data in Table E.1, the report could be as follows:
The eight participants were rank ordered by the magnitude of their difference scores and a Wilcoxon
T was used to evaluate the significance of the difference between treatments. The results
showed a significant difference, T 5 3, p < .05, with the positive ranks totaling 3 and the negative
ranks totaling 33.
■ Normal Approximation for the Wilcoxon T-Test
When a sample is relatively large, the values for the Wilcoxon T statistic tend to form a
normal distribution. In this situation, it is possible to perform the test using a z-score statistic
and the normal distribution rather than looking up a T value in the Wilcoxon table.
When the sample size is greater than 20, the normal approximation is very accurate and
can be used. For samples larger than n 5 50, the Wilcoxon table typically does not provide
any critical values, so the normal approximation must be used. The procedure for the
normal approximation for the Wilcoxon T is as follows:
1. Find the total for the positive ranks and the total for the negative ranks as before.
The Wilcoxon T is the smaller of the two values.
2. With n greater than 20, the Wilcoxon T values form a normal distribution with a
mean of
and a standard deviation of
m5
nsn 1 1d
4
nsn 1 1ds2n 1 1d
s5Î
24
The Wilcoxon T from the sample data corresponds to a z-score in this distribution
defined by
z 5 X 2m
s
5
nsn 1 1d
T 2
Î
4
nsn 1 1ds2n 1 1d
24
3. The unit normal table is used to determine the critical region for the z-score. For
example, the critical values are ±1.96 for a 5 .05.
Although the normal approximation is intended for samples with at least n 5 20
individuals, we will demonstrate the calculations with the data in Table E.1. These data
have n 5 8 and produced T 5 3. Using the normal approximation, these values produce
m5
nsn 1 1d
4
5 8s9d
4 5 18
nsn 1 1ds2n 1 1d
s5Î
24
5Î 8s9ds17d
24
5 Ï51 5 7.14
With these values, the obtained T 5 3 corresponds to a z-score of
z 5 T 2m
s
5 3 2 18
7.14 5 215
7.14 522.10