What this answers

This test answers "which specific pairs of groups actually differ?" after a Kruskal-Wallis omnibus test has already told you that some difference exists somewhere among three or more groups. A significant omnibus test never identifies which pairs are responsible on its own; that is exactly this test's job.

How it is calculated

Every observation across all groups is ranked together, exactly as in a Kruskal-Wallis test. For every pair of groups, the difference in mean rank is converted to a z score using a standard error based on the total sample size and each group's size, and a two-sided p-value follows from the normal distribution. Because testing many pairs inflates the chance of at least one false positive, this calculator applies Holm's step-down adjustment by default: p-values are sorted, adjusted progressively, and forced to be non-decreasing in that sorted order, a less conservative but equally valid alternative to the classic Bonferroni correction.

Worked example

For three clearly separated groups (1 to 3, 10 to 12, and 20 to 22), every pairwise comparison shows a large rank difference, and all three raw p-values are small. Holm's adjustment multiplies the smallest raw p-value by 3, the next by 2, and leaves the largest as-is, then enforces that the adjusted values never decrease going from most to least significant, typically leaving all three comparisons significant here given how cleanly separated the groups are.

Assumption audit

Calculated from your data: every pairwise mean rank difference, z score, raw p-value, and Holm-adjusted p-value across all group pairs.

Evidence to review: how many pairwise comparisons you are making; more groups mean more pairs, and more pairs mean the Holm correction pushes adjusted p-values higher, which is the intended, honest cost of testing more hypotheses at once.

You must verify: that a Kruskal-Wallis omnibus test on this same data was already significant; running pairwise comparisons without that first gate is a common source of inflated false-positive findings across a research literature.

What this result does not mean

An adjusted p-value below your threshold for one pair means that specific pair shows evidence of a difference after accounting for the other comparisons made, not that the difference is large; check the mean rank difference itself for a sense of magnitude, since rank-based p-values do not directly convey effect size.

Limitations

This calculator's per-pair standard error does not include the ranking tie-correction term some references add, a documented simplification. Holm's adjustment is applied only across the pairs entered in one calculation; running separate Dunn calculations in sequence for different subsets of groups does not achieve the same family-wise control as one calculation across all pairs at once.