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 697
Using the data in Table E.2(b), the Kruskal-Wallis formula produces a chi-square value of
H 5
15s16d1 12 39.52 1 26.52 1 542
5 5 5 2 2 3s16d
5 0.05(312.05 1 140.45 1 583.2) 2 48
5 0.05(1035.7) 2 48
5 51.785 2 48
5 3.785
With df 5 2, the chi-square table lists a critical value of 5.99 for a 5 .05. Because
the obtained chi-square value (3.785) is not greater than the critical value, our statistical
decision is to fail to reject H 0
. The data do not provide sufficient evidence to conclude that
there are significant differences among the three treatments.
As with the Mann-Whitney and the Wilcoxon tests, there is no standard format for
reporting the outcome of a Kruskal-Wallis test. However, the report should provide a summary
of the data, the value obtained for the chi-square statistic, as well as the value of df,
N, and the p value. For the Kruskal-Wallis test that we just completed, the results could
be reported as follows:
After ranking the individual scores, a Kruskal-Wallis test was used to evaluate differences among
the three treatments. The outcome of the test indicated no significant differences among the treatment
conditions, H 5 3.785 (2, N 5 15), p . .05.
There is one assumption for the Kruskal-Wallis test that is necessary to justify using the
chi-square distribution to identify critical values for H. Specifically, each of the treatment
conditions must contain at least five scores.
E.5 The Friedman Test: An Alternative
to the Repeated-Measures ANOVA
The Friedman test is used to evaluate the differences between three or more treatment
conditions using data from a repeated-measures design. This test is an alternative to the
repeated-measures ANOVA that was introduced in Chapter 13. However, the ANOVA
requires numerical scores that can be used to compute means and variances. The Friedman
test simply requires ordinal data. The Friedman test is also similar to the Wilcoxon test that
was introduced earlier in this chapter. However, the Wilcoxon test is limited to comparing
only two treatments, whereas the Friedman test is used to compare three or more treatments.
■ The Data for a Friedman Test
The Friedman test requires only one sample, with each individual participating in all of
the different treatment conditions. The treatment conditions must be rank-ordered for
each individual participant. For example, a researcher could observe a group of children
diagnosed with ADHD in three different environments: at home, at school, and during
unstructured play time. For each child, the researcher observes the degree to which the
disorder interferes with normal activity in each environment, and then ranks the three
environments from most disruptive to least disruptive. In this case, the ranks are obtained
by comparing the individual’s behavior across the three conditions. It is also possible for
each individual to produce his or her own rankings. For example, each individual could be