Kruskal-Wallis Test Calculator

The Kruskal-Wallis test is a non-parametric statistical test used to determine whether there are statistically significant differences between two or more independent groups. It is the non-parametric equivalent of one-way ANOVA and works by ranking all observations together, then comparing mean ranks across groups. It does not assume a normal distribution.

The test assumes that observations are independent across groups, the dependent variable is at least ordinal, and the groups have similarly shaped distributions (though they don't need to be normal). Unlike ANOVA, it does not require equal variances or normality.

All observations across all groups are pooled and ranked. The H statistic is calculated as H = (12 / N(N+1)) × Σ(Rᵢ² / nᵢ) − 3(N+1), where N is the total sample size, Rᵢ is the sum of ranks in group i, and nᵢ is the sample size of group i. H follows a chi-squared distribution with k−1 degrees of freedom.

Use Kruskal-Wallis when your data violates the normality assumption required by ANOVA, when you have ordinal data (e.g. Likert scale responses), when sample sizes are small, or when outliers are present that you cannot remove. It is a robust alternative that sacrifices some statistical power in exchange for fewer assumptions.

A significant result (p < α) indicates that at least one group's distribution differs from the others. It does not tell you which specific groups differ. To identify which pairs differ, you should follow up with a post-hoc test such as Dunn's test with a correction method like Bonferroni or Sidak.

The test works with small samples, but having at least 5 observations per group is generally recommended for the chi-squared approximation to be reliable. Very small groups (n < 5) may require exact p-value computation rather than the chi-squared approximation.

If the p-value is less than your chosen significance level (α, typically 0.05), you reject the null hypothesis and conclude that at least one group is statistically different. If p ≥ α, you fail to reject the null hypothesis and conclude there is insufficient evidence of a difference between groups.

Mann-Whitney U is used to compare exactly two independent groups, while Kruskal-Wallis generalizes this to three or more groups. In fact, when only two groups are provided, the Kruskal-Wallis test produces results equivalent to the Mann-Whitney U test with normal approximation.