As an aside, for a good and philosophically rigorous criticism of cavalier assumptions of normality or (arguably) pseudo-explanations that involve the central limit theorem, I’d recommend Lyon (2014), Why are Normal Distributions Normal?
Basically I think that whenever we are in the business of understanding how things actually work/”why” we’re seeing the data distributions we’re seeing, often-invoked explanations like the CLT or “multiplicative” CLT are kind of the tip of the iceberg that provides the “actual” explanation (rather then being literally correct by themselves), this iceberg having to do with the principle of maximum entropy / the tendency for entropy to increase / ‘universality’ and the fact that certain types of distributions are ‘attractors’ for a wide range of generating processes. I’m too much of an ‘abstract algebra person’ to have a clear sense of what’s going on, but I think it’s fairly clear that the folk story of “a lot of things ‘are’ normally distributed because of ‘the’ central limit theorem” is at best an ‘approximation’ and at worst misleading.
(One ‘mathematical’ way to see this is that it’s fishy that there are so many different versions of the CLT rather than one clear ‘canonical’ or ‘maximally general’ one. I guess stuff like this also is why I tend to find common introductions to statistics horribly unaesthetic and have had a hard time engaging with them.)
As an aside, for a good and philosophically rigorous criticism of cavalier assumptions of normality or (arguably) pseudo-explanations that involve the central limit theorem, I’d recommend Lyon (2014), Why are Normal Distributions Normal?
Basically I think that whenever we are in the business of understanding how things actually work/”why” we’re seeing the data distributions we’re seeing, often-invoked explanations like the CLT or “multiplicative” CLT are kind of the tip of the iceberg that provides the “actual” explanation (rather then being literally correct by themselves), this iceberg having to do with the principle of maximum entropy / the tendency for entropy to increase / ‘universality’ and the fact that certain types of distributions are ‘attractors’ for a wide range of generating processes. I’m too much of an ‘abstract algebra person’ to have a clear sense of what’s going on, but I think it’s fairly clear that the folk story of “a lot of things ‘are’ normally distributed because of ‘the’ central limit theorem” is at best an ‘approximation’ and at worst misleading.
(One ‘mathematical’ way to see this is that it’s fishy that there are so many different versions of the CLT rather than one clear ‘canonical’ or ‘maximally general’ one. I guess stuff like this also is why I tend to find common introductions to statistics horribly unaesthetic and have had a hard time engaging with them.)