The Theorem That Powers Statistics
The Central Limit Theorem (CLT) is the reason most of statistics works. It says that no matter what distribution you start with — uniform, exponential, bimodal, or anything else — the average of many independent samples will follow a normal distribution. This universality is what makes polls reliable, clinical trials valid, and quality control possible.
Watching Normality Emerge
This simulation lets you see the CLT in action. Choose a wildly non-normal source distribution (try exponential or bimodal), then increase the sample size. With sample size 1, the distribution of means matches the source. By sample size 5-10, it starts looking bell-shaped. By 30, it is nearly perfectly Gaussian. The transformation is startling and beautiful.
The Standard Error
The CLT also tells us how spread out the sample means will be: the standard error equals the population standard deviation divided by the square root of the sample size. This explains why averaging more data gives better estimates — but with diminishing returns. To cut the error in half, you need four times as much data.
History and Importance
The CLT was first proved in limited form by Abraham de Moivre in 1733 and generalized by Pierre-Simon Laplace in 1812. The modern rigorous version was established by Aleksandr Lyapunov in 1901. Today it underpins virtually all of inferential statistics, from the humble confidence interval to sophisticated Bayesian methods that assume approximate normality of posteriors.