Kruskal-Wallis Test Calculator — Rank ANOVA

The Kruskal-Wallis Test. Several independent groups may differ even when raw means are fragile. The Kruskal Wallis test replaces values with pooled ranks and asks whether group rank positions separate. Independence is the design. Every person or unit belongs to one group only, and observations must remain independent both within and between groups. Pool every observation while keeping its group label. Rank the combined values from smallest to largest on one shared rail, never separately inside each group. Equal pooled values receive their average rank. These midranks preserve ties without inventing an arbitrary ordering, and each tied block also contributes to the correction. Return the pooled ranks to their original groups. Each group now has a rank sum and a mean rank that describe where its observations sit in the common ordering. The H formula increases when group rank sums spread apart. It scales each squared rank sum by its group size, then subtracts the common rank baseline. Ties reduce the available rank variation. Divide the uncorrected H by the standard tie factor; if every value is identical, that factor is zero and inference stops.

This page compares H with the upper tail of a chi square distribution using the number of groups minus one degrees of freedom. Larger H is stronger evidence. That chi square reference is an approximation. Any group below five observations triggers a warning, but the calculator never switches automatically to an exact or permutation method. The sharp null says every group shares one distribution. Rejection means at least one rank distribution differs, but it does not identify which group caused the omnibus result. A median or location interpretation needs reasonably similar group shapes. Unequal spread or shape can also move pooled ranks, so check distributions before naming a median shift. Meaningful ordering is enough for the rank calculation. Normality and equal raw spacing are not required, but rank robustness does not rescue dependent or wrongly grouped data. The result table shows mean ranks, not raw means or medians. A high mean rank means observations tend to appear later in the pooled ordering, nothing more specific. The page reports eta squared H as a rank effect, floored at zero. It is not raw outcome variance explained, and this page provides no confidence interval.

Now compare three independent groups with five, four, and five observations. Keep alpha point zero five fixed and select Dunn Bonferroni as a possible follow up. Pool all fourteen values. There are no ties, so ranks one through fourteen stay ordinary and the tie correction equals one. The group rank sums are thirty-six, thirty-six, and thirty-three. Dividing by group size gives mean ranks seven point two, nine point zero, and six point six. Insert those rank sums and sizes into the formula. H equals zero point seven seven one four, far smaller than a strongly separated rank pattern. With two degrees of freedom, the upper chi square p-value is zero point six seven nine nine six five. It exceeds alpha point zero five. Fail to reject the common distribution null. Dunn rows do not run after this non-significant omnibus, and the floored rank effect is zero. This is not proof that the three populations are identical. It means these fourteen observations do not separate the group rank positions beyond routine chance variation.

For contrast, three perfectly separated groups have mean ranks three, eight, and thirteen. H equals twelve point five and p equals zero point zero zero one nine three. A significant omnibus opens the selected Dunn follow up. Each pair compares two mean ranks with a tie adjusted pooled rank standard error and a two-sided normal z probability. Three groups create three comparisons. Bonferroni multiplies every raw p-value by three and caps at one; here only group one versus group three remains significant. An omnibus rejection does not require every adjusted pair to reject. Report the global result and every adjusted row honestly instead of forcing a story that all groups differ. Tied values receive midranks and alter H through the tie factor. If all values are identical, or a group has fewer than two observations, validation blocks the result. Report group sizes, independence, pooled midranks, tie correction, H, degrees of freedom, chi square approximation, p, alpha, rank effect, warning, and adjusted follow ups. Use Friedman for repeated measures. Turn several independent groups into one honest rank comparison, guard the omnibus before pairwise claims, and run the Kruskal Wallis calculator free at Distri Scope dot com.