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 699
calculations for the Friedman test require the number of individuals in the sample (n) and
the number of treatment conditions (k). For the data in Table E.3(b), n 5 5 and k 5 4. The
Friedman test evaluates the differences between treatments by computing the following
test statistic:
x 2 r 5 12
nksk 1 1d (oR)2 2 3nsk 1 1d
Note that the statistic is identified as chi-square (x 2 ) with a subscript r, and corresponds
to a chi-square statistic for ranks. This chi-square statistic has degrees of freedom determined
by df 5 k 2 1, and is evaluated using the critical values in the chi-square distribution
shown in Table B8 in Appendix B.
For the data in Table E.3(b), the statistic is
x 2 r 5 12
5s4ds5d s62 1 9.5 2 1 15.5 2 1 19 2 d 2 3s5ds5d
5 12 s36 1 90.25 1 240.25 1 361d 2 75
100
5 0.12(727.5) 2 75
5 12.3
With df 5 k 2 1 5 3, the critical value of chi-square is 7.81. Therefore, the decision is
to reject the null hypothesis and conclude that there are significant differences among the
four treatment conditions.
As with most of the tests for ordinal data, there is no standard format for reporting the
results from a Friedman test. However, the report should provide the value obtained for
the chi-square statistic as well as the values for df, n, and p. For the data that were used to
demonstrate the Friedman test, the results would be reported as follows:
After ranking the original scores, a Friedman test was used to evaluate the differences among the
four treatment conditions. The outcome indicated that there are significant differences, x 2 r 5 12.3
(3, n 5 5), p , .05.