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

688 APPENDIX E | Hypothesis Tests for Ordinal Data: Mann-Whitney, Wilcoxon, Kruskal-Wallis, and Friedman Tests
3. The original scores may have unusually high variance. Variance is a major component
of the standard error in the denominator of t statistics and the error term in
the denominator of F-ratios. Thus, large variance can greatly reduce the likelihood
that these parametric tests will find significant differences. Converting the scores to
ranks essentially eliminates the variance. For example, 10 scores have ranks from 1
to 10 no matter how variable the original scores are.
4. Occasionally, an experiment produces an undetermined, or infinite, score. For
example, a rat may show no sign of solving a particular maze after hundreds of
trials. This animal has an infinite, or undetermined, score. Although there is no
absolute score that can be assigned, you can say that this rat has the highest score
for the sample and then rank the rest of the scores by their numerical values.
■ Ranking Tied Scores
Whenever you are transforming numerical scores into ranks, you may find two or more
scores that have exactly the same value. Because the scores are tied, the transformation
process should produce ranks that are also tied. The procedure for converting tied scores
into tied ranks was presented in Chapter 15 (page 513) when we introduced the Spearman
correlation, and is repeated briefly here. First, you list the scores in order, including tied
values. Second, you assign each position in the list a rank (1st, 2nd, and so on). Finally,
for any scores that are tied, you compute the mean of the tied ranks, and use the mean
value as the final rank. The following set of scores demonstrates this process for a set of
n 5 8 scores.
Original scores: 3 4 4 7 9 9 9 12
Position ranks: 1 2 3 4 5 6 7 8
Final ranks: 1 2.5 2.5 4 6 6 6 8
■ Statistics for Ordinal Data
You should recall from Chapter 1 that ordinal values tell you only the direction from one
score to another, but provide no information about the distance between scores. Thus,
you know that first place is better than second or third, but you do not know how much
better. Because the concept of distance is not well defined with ordinal data, it generally
is considered unwise to use traditional statistics such as t tests and analysis of variance
with scores consisting of ranks or ordered categories. Therefore, statisticians have developed
special techniques that are designed specifically for use with ordinal data.
In this chapter we introduce four hypothesis-testing procedures that are used with
ordinal data. Each of the new tests can be viewed as an alternative for a commonly used
parametric test. The four tests and the situations in which they are used are as follows:
1. The Mann-Whitney test uses data from two separate samples to evaluate the difference
between two treatment conditions or two populations. The Mann-Whitney
test can be viewed as an alternative to the independent-measures t hypothesis test
introduced in Chapter 10.
2. The Wilcoxon test uses data from a repeated-measures design to evaluate the
difference between two treatment conditions. This test is an alternative to the
repeated-measures t test from Chapter 11.
3. The Kruskal-Wallis test uses data from three or more separate samples to evaluate
the differences between three or more treatment conditions (or populations). The
Kruskal-Wallis test is an alternative to the single-factor, independent-measures
ANOVA introduced in Chapter 12.