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Central Limit Theorem

The central limit theorem is the reason the normal distribution is everywhere. It says that averaging washes out the shape of whatever you started with.

For independent, identically distributed X1,…,XnX_1, \ldots, X_n with mean μ\mu and finite variance σ2\sigma^2:

The central limit theorem
Xˉn−μσ/n⟶dN(0,1)\frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} \overset{d}{\longrightarrow} \mathcal{N}(0,1)

Standardise the sample mean and its distribution goes normal, whatever you started from.

The individual XiX_i can be uniform, exponential, or a lumpy discrete mess. Average enough of them and the average is normal.

Start at n=1n = 1, where the histogram is the population itself and the overlaid curve fits badly, then walk nn up. The coin's two spikes become a bell within about ten draws. The exponential takes far longer and leans right well past the usual threshold, which is why "how large is large enough" below has no single answer.

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