Rank test (no normal distribution)

Mann-Whitney (two groups), Kruskal-Wallis (three or more groups) and Wilcoxon (two measurements of the same people) order all values by size and work with rank positions instead of means. They do not need a normal distribution and are hardly disturbed by single outliers. As a size in the unit of measurement, the Hodges-Lehmann estimate (HL) with 95% confidence interval is given: for two groups the median of all differences between one person from each group, for two measurements the pseudo-median of the change. The effect size is r = |z| / √N (.1 small, .3 medium, .5 large), the rank-biserial correlation (share of pairs in which one group is higher, times 2, minus 1) or, for Kruskal-Wallis, ε² = H / (n − 1). After a significant Kruskal-Wallis test, the Dunn test with Holm adjustment shows which pairs differ. Statable names a reason for the rank test before the calculation: clear skewness, outliers or a measure with few levels (ordinal or at most 7 distinct values) - group size alone is not one. Statable never switches by itself.

Example: Ozone values in May against August: U = 127.5, z = 3.85, p < .001, r = .53 - the values in August are clearly higher.

Caution: A rank test shows whether one group tends to have higher values; it does not test the medians alone. With very many tied values and very small groups the normal approximation is imprecise.