Although one-way analysis of variance (ANOVA) is the method of choice when testing for differences between multiple groups, it assumes that the mean is a valid estimate of center and that the distribution of the test variable is reasonably normal and similar in all groups. However, when your test variable is ordinal, the mean is not a valid estimate because the distances between the values are arbitrary. Even if the mean is valid, the distribution of the test variable may be so non-normal that it makes you suspicious of any test that assumes normality.

When the assumptions behind the standard ANOVA are invalid or suspect, you should consider using the nonparametric procedures designed to test for the significance of the difference between multiple groups. They are called nonparametric because they make no assumptions about the parameters (such as the mean and variance) of a distribution, nor do they assume that any particular distribution is being used. In this chapter, we discuss two nonparametric tests for multiple independent samples, the Kruskal-Wallis and median tests.

The median method tests the null hypothesis that two or more independent samples have the same median. It assumes nothing about the distribution of the test variable, making it a good choice when you suspect that the distribution varies by group.

The Kruskal-Wallis test is a one-way analysis of variance by ranks. It tests the null hypothesis that multiple independent samples come from the same population. Unlike standard ANOVA, it does not assume normality, and it can be used to test ordinal variables.

In addition to their standard output, both the Kruskal-Wallis and median tests will display descriptive statistics and/or quartiles of the test variable.