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

APPENDIX E | Hypothesis Tests for Ordinal Data: Mann-Whitney, Wilcoxon, Kruskal-Wallis, and Friedman Tests 689
4. The Friedman test uses data from a repeated-measures design to compare the differences
between three or more treatment conditions. This test is an alternative to
the repeated-measures ANOVA from Chapter 13.
In each case, you should realize that the new, ordinal-data tests are back-up procedures
that are available in situations in which the standard, parametric tests cannot be used. In
general, if the data are appropriate for an ANOVA or one of the t tests, then the standard
test is preferred to its ordinal-data alternative.
E.2 The Mann-Whitney U-Test: An Alternative
to the Independent-Measures t Test
Recall that a study using two separate samples is called an independent-measures study
or a between-subjects study. The Mann-Whitney test is designed to use the data from two
separate samples to evaluate the difference between two treatments (or two populations).
The calculations for this test require that the individual scores in the two samples be rankordered.
The mathematics of the Mann-Whitney test are based on the following simple
observation:
A real difference between the two treatments should cause the scores in one sample to be
generally larger than the scores in the other sample. If the two samples are combined and
all the scores are ranked, then the larger ranks should be concentrated in one sample and the
smaller ranks should be concentrated in the other sample.
■ The Null Hypothesis for the Mann-Whitney Test
Because the Mann-Whitney test compares two distributions (rather than two means), the
hypotheses tend to be somewhat vague. We state the hypotheses in terms of a consistent,
systematic difference between the two treatments being compared.
H 0
: There is no difference between the two treatments. Therefore, there is no
tendency for the ranks in one treatment condition to be systematically higher
(or lower) than the ranks in the other treatment condition.
H 1
: There is a difference between the two treatments. Therefore, the ranks in one
treatment condition are systematically higher (or lower) than the ranks in the
other treatment condition.
■ Calculating the Mann-Whitney U
Again, we begin by combining all the individuals from the two samples and then rank
ordering the entire set. If you have a numerical score for each individual, combine the
two sets of scores and rank order them. The Mann-Whitney U is then computed as if
the two samples were two teams of athletes competing in a sports event. Each individual
in sample A (the A team) gets one point whenever he or she is ranked ahead of
an individual from sample B. The total number of points accumulated for sample A is
called U A
. In the same way, a U value, or team total, is computed for sample B. The
final Mann-Whitney U is the smaller of these two values. This process is demonstrated
in the following example.