Central limit theorem demo
The most important theorem in statistics, watched rather than proved. Pick a lopsided parent distribution, slide the sample size up, and see the histogram of sample means pulled into the exact normal curve the theorem predicts.
4,000 sample means of size n = 5
Parent skewness γ₁ = 2.00 · sample-mean skewness γ₁/√n ≈ 0.89
The gold curve is N(μ, σ²/n) — the CLT’s prediction, with nothing fitted. At n = 1 the histogram is just the parent (spikes and all); by n = 30 even the skewed parents look respectably normal, and the axis labels show the standard error shrinking at rate 1/√n. The mean’s skewness dies as γ₁/√n, so to slow convergence down, give it more γ₁: drop the gamma shape α below 1 (α = 1 is the exponential), push the lognormal σ towards 1.5, or the Bernoulli p towards 0.02, and the bell takes far longer to arrive. Uniform has no dial because it has no skew to give — it is the symmetric baseline the others are compared against. Bernoulli is the fun one: two spikes become a binomial comb and then a bell.
Why examiners never stop asking it
The CLT is why the normal distribution runs statistics: whatever the parent — discrete, skewed, bounded — the standardised sample mean tends to N(0, 1), so X̄ is approximately N(μ, σ²/n) once n is moderate. That single fact powers confidence intervals, hypothesis tests, the normal approximations to the binomial and Poisson, and half of CS1’s sampling theory. The demo makes the two practical caveats visible too: convergence is slower for heavily skewed parents (drop the gamma shape α below 1, push the lognormal σ up, or the Bernoulli p down, and compare with uniform at n = 10), and nothing about the parent needs to be normal — only its variance needs to exist.
Make it stick. The sampling distributions this feeds (t, chi-squared and F) live in our statistical tables, and every parent here is on the distributions cheat sheet. Memori is a flashcard app built by actuarial students, with a ready-made CS1 set in the shop. Join the beta.
For education only.