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.