Recently, I was looking at some slides from a data science course, and one statement was presented rather matter-of-factly:
The normal distribution is often a good model for variation in natural phenomena.
That caught me off guard and sent me down a rabbit hole into probability theory and the Central Limit Theorem. I think I have a decent intuitive grasp of why the CLT works, so I don’t necessarily need a full proof (though I wouldn’t mind one). What I’m really trying to understand is why it’s considered so significant.
Yes, the theorem tells us that the sampling distribution of the mean tends toward normality but why is that such a big deal? It feels like we’re shifting the focus to averages rather than addressing the underlying population directly. We can make statements about the mean, but that seems somewhat limited. It almost feels like we’re reframing—if not avoiding—the original question we care about.
The ingredient you might be missing for how common the CLT is applicable is the following: in most complex systems (e.g. biological systems), any variable you measure is likely influenced by a lot of other hidden variables. Because there are so many variables at play, each effect is likely to be small, and the way their effects are compounded is likely somewhat additive (this one comes from things like series expansion). Hence summing up effects between variables must be relatively common and account for the bulk of the variation in a response variable.
A last bit is this: most statistical methods like the linear model are relatively robust to deviation from the normal distribution. So, you don't need exactly a normal distribution, you just need close enough. It turns out the CLT often produces "close enough" quite quickly (i.e. with a few variables added together).