Math

Why the bell appears

Add up draws from any shape - a die, a loaded coin, two humps, the long tail of a waiting time - and the standardised sum turns into the same bell. One sample is collected before your eyes and becomes one point of the series beside it; the sum, the mean and the standardised sum are three names of that point. Cumulants of a sum add, so the skewness falls as 1/√n and the excess kurtosis as 1/n, and the sheet measures both. And one source breaks it: a Cauchy has no variance, and no bell ever appears.

By Mykhailo Makarov

Actual application image

What happens

Add up many small independent pieces and the sum falls on the same bell curve whatever the pieces were - that is the central limit theorem, and this module shows it happening and measures how fast. Skewness of the sum dies as 1/√n, excess kurtosis as 1/n: a coin's sum is bell-shaped by a dozen throws, an exponential's lopsided tail needs many more. The exact distribution of the sum is built by convolution and compared with a sampled one and with the normal. Pick a source, raise n, and watch the histogram straighten while the Kolmogorov distance falls.

Equations used in the module

Equations
Z = (X₁ + … + X_n − nμ) / (σ√n)
κ_k(Z) = κ_k(X) / (σ^k n^{k/2−1}):  skewness ∝ 1/√n, excess kurtosis ∝ 1/n
uniform: γ₂ = −6/5;  exponential: γ₁ = 2, γ₂ = 6
Kolmogorov distance to the normal, exact pmf by convolution

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