# Kruskal-Wallis test

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## **Background**

In statistics, the Kruskal–Wallis one-way analysis of variance by ranks (named after William Kruskal and W. Allen Wallis) is a non-parametric method for testing whether samples originate from the same distribution. It is used for comparing more than two samples that are independent, or not related. The parametric equivalence of the Kruskal-Wallis test is the one-way analysis of variance (ANOVA). The factual null hypothesis is that the populations from which the samples originate have the same median. When the Kruskal-Wallis test leads to significant results, then at least one of the samples is different from the other samples. The test does not identify where the differences occur or how many differences actually occur. It is an extension of the Mann–Whitney U test to 3 or more groups. The Mann-Whitney would help analyze the specific sample pairs for significant differences.

Since it is a non-parametric method, the Kruskal–Wallis test does not assume a normal distribution, unlike the analogous one-way analysis of variance. However, the test does assume an identically shaped and scaled distribution for each group, except for any difference in medians. Kruskal–Wallis is also used when the groups being examined are of unequal size (different number of participants).

## **Additional information**

A large amount of computing resources are required to compute exact probabilities for the Kruskal-Wallis test. Existing software only provides exact probabilities for sample sizes less than about 30 participants. These software programs rely on asymptotic approximation for larger sample sizes. Exact probability values for larger sample sizes are available. Spurrier (2003) published exact probability tables for samples as large as 45 participants. Meyer and Seaman (2006) produced exact probability distributions for samples as large as 105 participants.

If the statistic is not significant, then there is no evidence of differences between the samples. However, if the test is significant then a difference exists between at least two of the samples. Therefore, a researcher might use sample contrasts between individual sample pairs, or post hoc tests, to determine which of the sample pairs are significantly different. When performing multiple sample contrasts, the Type I error rate tends to become inflated.

## **Notes & References**

[1] Kruskal and Wallis (1952) "Use of ranks in one-criterion variance analysis", Journal of the American Statistical Association 47 (260): 583–621.

[2] Spurrier, J. D. (2003). "On the null distribution of the Kruskal-Wallis statistic". Journal of Nonparametric Statistics, 15(6), 685-691.

## **Credits & Notices**

* Authors-contributors* to this page (listed alphabetically, last name, first & middle initial only, no institutional affiliations, no scientific titles):

Stawicki SP

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