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

692 APPENDIX E | Hypothesis Tests for Ordinal Data: Mann-Whitney, Wilcoxon, Kruskal-Wallis, and Friedman Tests
compared two treatments, A and B, using a separate sample of n 5 6 for each treatment.
The data produced a value of U 5 6. The z-score corresponding to U 5 6 is
z 5
5
U 2 n n A B
Î
2
n n sn 1 n
A B A B
1 1d
12
6 2 6s6d
Î
2
6s6ds6 1 6 1 1d
12
5
Î 212
468
12
521.92
With a 5 .05, the critical value is z 5 61.96. Our computed z-score, z 5 21.92, is not in
the critical region, so the decision is to fail to reject H 0
. Note that we reached the same conclusion
for the original test using the critical values in the Mann-Whitney U table.
E.3 The Wilcoxon Signed-Ranks Test: An Alternative
to the Repeated-Measures t Test
The Wilcoxon test is designed to evaluate the difference between two treatments, using the
data from a repeated-measures experiment. Recall that a repeated-measures study involves
only one sample, with each individual in the sample being measured twice. The difference
between the two measurements for each individual is recorded as the score for that individual.
The Wilcoxon test requires that the differences be rank-ordered from smallest to largest in
terms of their absolute magnitude, without regard for sign or direction. For example, Table E.1
shows differences scores and ranks for a sample of n 5 8 participants.
TABLE E.1
Ranking difference scores.
Note that the differences
are ranked by magnitude,
independent of direction.
Participant
Difference from
Treatment 1 to Treatment 2
Rank
A 14 1
B 214 5
C 15 2
D 220 7
E 26 3
F 216 6
G 28 4
H 224 8