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
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. 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 (compare lognormal 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, 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.