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

696 APPENDIX E | Hypothesis Tests for Ordinal Data: Mann-Whitney, Wilcoxon, Kruskal-Wallis, and Friedman Tests
TABLE E.2
Preparing a set of data for
analysis using the Kruskal-
Wallis test. The original
data consisting of numerical
scores are shown in table
(a). The original scores are
combined into one group
and rank ordered using
the standard procedure for
ranking tied scores. The
ranks are then substituted
for the original scores to
create the set of ordinal
data shown in table (b).
(a) Original Numerical Scores
I II III
14 2 26 N 5 15
3 14 8
21 9 14
5 12 19
16 5 20
n 1
5 5 n 2
5 5 n 3
5 5
(b) Ordinal Data (Ranks)
I II III
9 1 15 N 5 15
2 9 5
14 6 9
3.5 7 12
11 3.5 13
T 1
5 39.5 T 2
5 26.5 T 3
5 54
n 1
5 5 n 2
5 5 n 3
5 5
■ The Null Hypothesis for the Kruskal-Wallis Test
As with the other tests for ordinal data, the null hypothesis for the Kruskal-Wallis test
tends to be somewhat vague. In general, the null hypothesis states that there are no differences
among the treatments being compared. Somewhat more specifically, H 0
states that
there is no tendency for the ranks in one treatment condition to be systematically higher (or
lower) than the ranks in any other condition. Generally, we use the concept of “systematic
differences” to phrase the statement of H 0
and H 1
. Thus, the hypotheses for the Kruskal-
Wallis test are phrased as follows:
H 0
: There is no tendency for the ranks in any treatment condition to be systematically
higher or lower than the ranks in any other treatment condition. There are
no differences between treatments.
H 1
: The ranks in at least one treatment condition are systematically higher (or
lower) than the ranks in another treatment condition. There are differences
between treatments.
Table E.2(b) presents the notation that is used in the Kruskal-Wallis formula along with
the ranks. The notation is relatively simple and involves the following values.
1. The ranks in each treatment are added to obtain a total or T value for that treatment
condition. The T values are used in the Kruskal-Wallis formula.
2. The number of subjects in each treatment condition is identified by a lowercase n.
3. The total number of subjects in the entire study is identified by an uppercase N.
The Kruskal-Wallis formula produces a statistic that is usually identified with the letter
H and has approximately the same distribution as chi-square, with degrees of freedom
defined by the number of treatment conditions minus one. For the data in Table E.2(b),
there are 3 treatment conditions, so the formula produces a chi-square value with df 5 2.
The formula for the Kruskal-Wallis statistic is
H 5
12
NsN 1 1d1 n2 ST2 2 3sN 1 1d