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Kruskal-Wallis Test

The Kruskal-Wallis Test is a non-parametric test that examines whether a sample originates from a particular distribution. Much like the ANOVA Test, it is a one-way analysis of variance by ranks. The actual test does not aim to detect the differences between populations being tested, but rather aims to quantify the degree of deviation between the two sets of sample populations – thus, generating a p-value statistic to test for significance.

Much like the Friedman Test, and unlike the ANOVA test, the Kruskal-Wallis Test does not assume a normal distribution of the sample population. However, it does assume that the two groups being tested have the same type of distribution.

The Kruskal-Wallis Test Statistic is commonly written in the following form:

K = (N-1)[( ∑_g_n_i(r_i-r)^2/∑_g∑_n_(r_ij-r)^2)



n_i is the number of observations in group i

r_ij is the rank of observation j from group I       

N is the total number of observations

r_i = ∑r_ij/n_i

= 1/2(N+1)


After deriving the value of K, one can look at distribution tables of the K-statistic to obtain the p-value. If the p-value is smaller than 0.05, the null hypothesis of equal population medians can be rejected. However, if the p-value is higher than 0.05, then one would fail to reject the null hypothesis at the 5% significance level. Thus, understanding the Kruskal-Wallis Test is crucial for examining differences in two sets of sample populations.

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