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I'm making a code in R that contains some parametric and non-parametric tests, like ANOVA and Kruskal-Wallis.

To know if I should use one or another I check the "normality" of the test sample. My question is the following: my sample has thousands of values (let's say, around 10000) so I checked the histogram, boxplot and used the ad.test to check if can be accepted the normality assumption. Since is a large sample ($n\ge30$) we should consider that the sample could be normal, even if the p-value is below the significance level (0.05), but if the outliers have a lot effect, we should reject the normality assumption, right? Is there any percentage of outliers that must exist to reject the normality hypothesis?

Sorry if this question is confusing. I'm not used to work on statistics so I'm a little confused with this topic.

Regards, Bernardo

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  • $\begingroup$ The distribution of a random variable has no relation with sample size. If the random variable does not follow normal, even the sample size is million level, it still is not normally distributed. $\endgroup$
    – user158565
    Commented Jul 24, 2019 at 23:23
  • $\begingroup$ Large samples don't change the shape of the population; you're confusing something about how the distribution of sample means tend to behave (though the specific threshold value of 30 is pretty much a bogus idea even there) with that of ordinary sampling. This is discussed in many answers (and probably many more comments) on site already. You will need to revise your question in light of the fact that the shape of the population distribution will be whatever it is and abandon the premise that taking large samples will make it any different. $\endgroup$
    – Glen_b
    Commented Jul 25, 2019 at 0:33
  • $\begingroup$ Thank you for your answers. So basically I only need to do the ad.test and conclude that if the normality assumption can be accepted or not. Is that correct? $\endgroup$ Commented Jul 25, 2019 at 2:09
  • $\begingroup$ stats.stackexchange.com/a/2501/11849 $\endgroup$
    – Roland
    Commented Jul 25, 2019 at 4:31

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The shape of the distribution does not depend on sample size, but the p value does. That is, with a large sample the result of ad test (or any test) of normality will give a low p value for a minor deviation from normality. So, I would not use ad test at all.

Also, outliers, influential points and so on are often problematic for ANOVA/OLS regression. What I would do is run a model that deals with outliers such as robust regression or quantile regression. You could then compare results with OLS regression and see if it makes a big difference.

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