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

690 APPENDIX E | Hypothesis Tests for Ordinal Data: Mann-Whitney, Wilcoxon, Kruskal-Wallis, and Friedman Tests
EXAMPLE E.1
We begin with two separate samples with n 5 6 scores in each.
Sample A (treatment 1): 27 2 9 48 6 15
Sample B (treatment 2): 71 63 18 68 94 8
Next, the two samples are combined, and all 12 scores are ranked.
Ranks for Sample A 7 1 4 8 2 5
Ranks for Sample B 11 9 6 10 12 3
Each individual in sample A is assigned 1 point for every score in sample B that
has a higher rank. U A
5 4 1 6 1 5 1 4 1 6 1 5 5 30. Similarly, U B
5 0 1 0 1 2 1
0 1 0 1 4 5 6.
Thus, the Mann-Whitney U is 6. As a simple check on your arithmetic, note that
U A
1 U B
5 n A
n B
. For these data, 30 1 6 5 6(6).
■
■ Computing U for Large Samples
Because the process of counting points to determine the Mann-Whitney U can be
tedious, especially with large samples, there is a formula that will generate the U value
for each sample. To use this formula, you combine the samples and rank-order all
the individuals as before. Then you must find SR A
, which is the sum of the ranks for
individuals in sample A, and the corresponding SR B
for sample B. The U value for each
sample is then computed as follows: For sample A,
and for sample B,
U A
5 n A
n B
1 n A sn A 1 1d
2
2SR A
U B
5 n A
n B
1 n sn 1 1d
B B
2SR
2
B
These formulas are demonstrated using the data from Example E.1. For sample A, the
sum of the ranks is
For sample B, the sum of the ranks is
SR A
5 1 1 2 1 4 1 5 1 7 1 8 5 27
SR B
5 3 1 6 1 9 1 10 1 11 1 12 5 51
By using the special formula, for sample A,
U A
5 n A
n B
1 n sn 1 1d
A A
2
5 6s6d 1 6s7d
2 2 27
5 36 1 21 2 27
5 30
2SR A
For sample B,
U B
5 n A
n B
1 n sn 1 1d
B B
2
5 6s6d 1 6s7d
2 2 51
5 36 1 21 2 51
5 6
2SR B