Kruskal-Wallis Test

The Kruskal-Wallis test is a nonparametric (distribution free) test, and is used when the assumptions of one-way ANOVA are not met.  Both the Kruskal-Wallis test and one-way ANOVA assess for significant differences on a continuous dependent variable by a categorical independent variable (with two or more groups).  In the ANOVA, we assume that the dependent variable is normally distributed and there is approximately equal variance on the scores across groups.  However, when using the Kruskal-Wallis Test, we do not have to make any of these assumptions.  Therefore, the Kruskal-Wallis test can be used for both continuous and ordinal-level dependent variables.  However, like most non-parametric tests, the Kruskal-Wallis Test is not as powerful as the ANOVA.

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Null hypothesis: Null hypothesis assumes that the samples (groups) are from identical populations.

Alternative hypothesis: Alternative hypothesis assumes that at least one of the samples (groups) comes from a different population than the others.

Example questions answered:

How do test scores differ between the different grade levels in elementary school?

Do job satisfaction scores differ by race?

The distribution of the Kruskal-Wallis test statistic approximates a chi-square distribution, with k-1 degrees of freedom, if the number of observations in each group is 5 or more.  If the calculated value of the Kruskal-Wallis test is less than the critical chi-square value, then the null hypothesis cannot be rejecedt.  If the calculated value of Kruskal-Wallis test is greater than the critical chi-square value, then we can reject the null hypothesis and say that at least one of the samples comes from a different population.


  1. We assume that the samples drawn from the population are random.
    2. We also assume that the observations are independent of each other.
    3. The measurement scale for the dependent variable should be at least ordinal.


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