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From here I read that

The t-test assumes that the means of the different samples are normally distributed; it does not assume that the population is normally distributed. By the central limit theorem, means of samples from a population with finite variance approach a normal distribution regardless of the distribution of the population ...... t-test is invalid for small samples from non-normal distributions, but it is valid for large samples from non-normal distributions.

So my follow up question is, how do you know if the means are normally distributed when population is believed to be non-normal?

It basically says that if sample size is large enough, you can assume normality of the statistic of your interest, regardless of the shape of the actual population. But how do you know what the threshold of "large enough" is?

I'm interested in checking normality of both means and variances.

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  • $\begingroup$ The quoted section is wrong: The $t$-test does assume normally distributed populations. For a nice summary, see here. $\endgroup$
    – COOLSerdash
    Jun 23, 2019 at 8:17
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    $\begingroup$ You ask three questions and they differ sufficiently that you ought to tell us which is the one you want us to answer. The first about normal distribution of means is a technical question with a simple answer that's unlikely to be helpful for the other two; the question about knowing the threshold of "large enough" has an answer hinted at by @glen_b in his post in the linked thread; and the issue of checking normality appears to be a completely different one that is so general its answer likely has little practical relevance. $\endgroup$
    – whuber ♦
    Jun 23, 2019 at 11:45
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    $\begingroup$ Also your question is not fully motivated. Why do you care? Why do you think that samples have enough information to reliably inform us about normality? Why are you avoiding distribution-free (nonparametric) methods that are really efficient even under normality? $\endgroup$
    – Frank Harrell
    Jun 23, 2019 at 11:50

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