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I have a sample size that is more than 30 but one of my continuous variables is not normally distributed. The Shapiro-Willk test does say that my data is not normally distributed. I am going to compare the results of the subjects among a categorical variable with more than two levels and want to see if there are significant differences among the categorical levels. From my understanding the central limit theorem states that in even though it is not normally distributed, at a large enough sample size it would be normally distributed. The subjects only have one score (i.e. no pre and post score or test, just one score). Would greatly appreciate help in clarifying this. Thanks.

Yes I am aware that the Kruskal-Wallis test is for medians and not means as in ANOVA.I am also assuming that the sample is randomly chosen.

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    $\begingroup$ When you say "my data is not normally distributed," does that mean over all the data or within each category? if there are differences in your continuous variable among the categories, you wouldn't expect an overall normal distribution of the continuous variable. The assumptions of ANOVA significance testing are about normality of the continuous variable around its mean value within each category. See the page Is normality testing essentially useless? and its links for more discussion. $\endgroup$
    – EdM
    Commented Feb 27, 2022 at 20:47
  • $\begingroup$ hi @EdM, I meant for that specific category (test score in my case) and my three categories is the program of study are what I am seeing if there are differences among the test scores. Also that was what the Shapiro-Willk test told me for that specific variable $\endgroup$
    – ineedhelp
    Commented Feb 27, 2022 at 20:54
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    $\begingroup$ There's no one answer, too much depends on details of your data and your hypotheses. See this page, this page, this page, this page, among many others, and their links, for issues to consider and ways to proceed. $\endgroup$
    – EdM
    Commented Feb 28, 2022 at 14:00

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