# Kruskal-Wallis test

The Kruskal-Wallis test is used to test the null hypothesis that multiple population distribution functions (corresponding to multiple samples) are identical against the alternative hypothesis that they differ by location.

(For two samples, the Kruskal-Wallis test is equivalent to the two-sample rank-sum test.)

### Assumptions:

• Within each sample, the values are independent, and identically distributed. (The Kruskal-Wallis test is a nonparametric test. We need not specify or know what the distribution is, only that all the values in each sample follow the same continuous distribution.)
• The samples are independent of each other.
• The populations from which the different samples were taken differ only in location. That is, the populations may differ in their means or medians, but not in their dispersions or distributional shape (such as skewness).
• For multiple comparisons to be meaningful, the treatment effect is viewed as fixed, so that the populations (treatment groups) in the experiment include all those of interest.
• Because the test statistic for the Kruskal-Wallis test is based only on the ranks of the data values, the test can be performed when the only data available are those relative ranks.

### Guidance:

• Ways to detect before performing the Kruskal-Wallis test whether your data violate any assumptions.
• Ways to examine Kruskal-Wallis test results to detect assumption violations.
• Possible alternatives if your data or Kruskal-Wallis test results indicate assumption violations.

To properly analyze and interpret results of the Kruskal-Wallis test, you should be familiar with the following terms and concepts:

If you are not familiar with these terms and concepts, you are advised to consult with a statistician. Failure to understand and properly apply the Kruskal-Wallis test may result in drawing erroneous conclusions from your data. Additionally, you may want to consult the following references:

• Brownlee, K. A. 1965. Statistical Theory and Methodology in Science and Engineering. New York: John Wiley & Sons.
• Conover, W. J. 1980. Practical Nonparametric Statistics. 2nd ed. New York: John Wiley & Sons.
• Daniel, Wayne W. 1978. Applied Nonparametric Statistics. Boston: Houghton Mifflin.
• Daniel, Wayne W. 1995. Biostatistics. 6th ed. New York: John Wiley & Sons.
• Hollander, M. and Wolfe, D. A. 1973. Nonparametric Statistical Methods. New York: John Wiley & Sons.
• Lehmann, E. L. 1975. Nonparametrics: Statistical Methods Based on Ranks. San Francisco: Holden-Day.
• Miller, Rupert G. Jr. 1996. Beyond ANOVA, Basics of Applied Statistics. 2nd. ed. London: Chapman & Hall.
• Rosner, Bernard. 1995. Fundamentals of Biostatistics. 4th ed. Belmont, California: Duxbury Press.
• Sokal, Robert R. and Rohlf, F. James. 1995. Biometry. 3rd. ed. New York: W. H. Freeman and Co.
• Zar, Jerrold H. 1996. Biostatistical Analysis. 3rd ed. Upper Saddle River, NJ: Prentice-Hall.