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I need to compare more than 2 groups, each containing discrete data (more specifically, integer numbers between 1 and 9). Shapiro-Wilk tests come back as significant for most groups, meaning the data is--unsurprisingly--not normally distributed. My question is what is an appropriate statistical test to compare the different groups.

  • Since the data is not normally distributed, am I correct in avoiding Anova tests? Is there a suitable alternative that compares mean values?
  • Is the correct approach to use Kruskal-Wallis H tests (i.e. one-way ANOVA on ranks)?
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    $\begingroup$ In the information in your data actually quantitative? If not, is it at least ordinal? Or are 1 to 9 just category names? $\endgroup$
    – Christian Hennig
    Commented Mar 24, 2022 at 12:13
  • $\begingroup$ Nothing in life is normally distributed. Normal distribution based theory holds approximately for large enough samples for many distributions, and Shapiro-Wilks isn't a reliable indicator regarding whether normality-based methods can be used or not. For integers between 1 and 9 holding quantitative information standard ANOVA may be fine unless distributional shapes and/or variances are very different between groups. $\endgroup$
    – Christian Hennig
    Commented Mar 24, 2022 at 12:16
  • $\begingroup$ The data are whatever they are and should be treated as what they are instead of what I would like them to be, but if those $1$ through $9$ are just category names, that would simplify the problem. $\endgroup$
    – Dave
    Commented Mar 24, 2022 at 12:17
  • $\begingroup$ What do you mean "compare the different groups"? Do you want to know whether they have the same mean? Or do you want to know whether they are equally distributed? Or... $\endgroup$
    – frank
    Commented Mar 24, 2022 at 12:34

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