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  • $\begingroup$ As a small detail, chi-square tests to do with frequencies of categories have nothing to do with normal distributions. More importantly, the "for instance" in your first paragraph is really important and needs to be expanded. For example, generalized linear models allowing non-identity links and non-normal families work well for many problems. I'd say most data I work with are positively skewed and sometimes more awkward still, but that does not mean jumping to nonparametric tests. But such models don't usually feature in introductory texts or courses. $\endgroup$ Commented Aug 23, 2018 at 12:49