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I want to compare two populations that are:

  • Not normally distributed,
  • Do not follow the same distribution shape,
  • Have different sample sizes.

Is there any test which corresponds to these requirements? I have been looking into non-parametric tests, but all seem to either require the data to follow the same distribution shape, or have the same sample size.

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    $\begingroup$ What aspect of the distributions do you want to compare? Means? Medians? CDFs? $\endgroup$ – COOLSerdash May 6 at 15:31
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    $\begingroup$ Which nonparametric tests did you find that require the same sample size? // @COOLSerdash has the right idea. What do you want to compare? $\endgroup$ – Dave May 6 at 15:35
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    $\begingroup$ The Wilcoxon (Mann-Whitney U) test has no requirement that the groups have the same size. $\endgroup$ – Dave May 6 at 16:50
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    $\begingroup$ Mann-Whitney U does not require same-shape distributions. It only requires that for the interpretation to be about a shift in the location while leaving alone the rest of the distribution e.g., N(0,1) to N(1,1). That interpretation allows you to make the conclusion you want to make about the means, but when the distribution shapes are markedly different, wondering about a difference in means strikes me as unhelpful, given the many other differences. After all, N(1,1) and exp(1) have the same mean (and variance), yet those are pretty different distributions! $\endgroup$ – Dave May 6 at 17:51
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    $\begingroup$ As you may have realized by now, it is possible to speculate endlessly--without specific information about your data. Can you show sample sizes, data summaries, histograms, or boxplots of your data and be more specific about your goals? $\endgroup$ – BruceET May 6 at 21:03

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