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/* /* Neither Mann-Whitney nor Kruskal-Wallis is about medians in general */ new section */ new section
This is a common misunderstanding. First, to know, that the distributions vary *only* by location, the dispersions must be equal. If they are not, the difference can come from either location or dispersion and this is widely covered in literature. It's easy to make data, where the medians are equal to each other, yet the test rejects the null hypothesis - due to the dispersion. In this case the type significance level is not preserved at all. Second, these are pseudomedians, not medians. There are several cases: - when the distributions are of same shape and dispersion, and also symmetric, indeed it's about medians - when the distributions are of same dispersion, it's about pseudomedians - when no assumption can be made, then it's about stochastic equality.

And since the Kruskal-Wallis is a generalization of the Mann-Whitney to k samples, the same holds here.

A few references: "The Wilcoxon–Mann–Whitney Procedure Fails as a Test of Medians" - https://www.tandfonline.com/doi/abs/10.1080/00031305.2017.1305291 https://www.graphpad.com/guides/prism/latest/statistics/stat_nonparametric_tests_dont_compa.htm http://proc-x.com/2014/06/example-2014-6-comparing-medians-and-the-wilcoxon-rank-sum-test/ https://edisciplinas.usp.br/pluginfile.php/1065042/mod_resource/content/1/Mann%C2%ADWhitney%20test%20is%20not%20just%20a%20test.pdf https://stats.idre.ucla.edu/other/mult-pkg/faq/general/faq-why-is-the-mann-whitney-significant-when-the-medians-are-equal/