Central Limit Theorem Simulator

Central Limit Theorem

Sample size n 5

Draw speed 5 / frame

Samples drawn

Mean of means

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Observed SE

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Predicted SE

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How it works

Each tick draws n independent values from the selected source distribution — which can be sharply skewed (exponential) or two-humped (bimodal), nothing like a bell curve — and plots their average as one more entry in the growing histogram of sample means. The Central Limit Theorem says that histogram converges on a normal distribution regardless of the source's shape, narrowing around the true population mean as n grows. The cyan curve overlays the exact prediction so you can watch the bars settle onto it.

X̄ₙ = (X₁+X₂+...+Xₙ)/n
X̄ₙ → N(μ, σ²/n)   as n grows
SE = σ/√n

Sample size n — how many draws are averaged per sample; larger n narrows the histogram (SE shrinks as 1/√n) and makes it converge to normal faster, even from a lopsided source.
Source distribution — the population each individual draw comes from; its own shape barely matters once n is large enough.
Draw speed — how many new sample-means are generated per animation frame.
Reset histogram — clears all accumulated samples and re-derives the display range for the current n and source.

Real-world relevance: this is why polling averages, A/B test metrics and quality-control measurements can all be treated as approximately normal and analyzed with the same z/t-test machinery, even though the underlying individual measurements rarely look normal at all.

Sample-mean histogram Predicted N(μ, σ²/n)

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