Watch sample means converge to a normal distribution regardless of the population shape
Given a population with mean μ and standard deviation σ, the distribution of sample means X̄ approaches a normal distribution with mean μ and standard deviation σ/√n as n increases, regardless of the population shape.
The CLT is the foundation of statistical inference. It explains why the normal distribution appears so frequently in nature and justifies using z-tests, confidence intervals, and many other statistical procedures.
For most practical purposes, n ≥ 30 is sufficient for the CLT to provide a good normal approximation. However, the required n depends on how skewed the population is: symmetric distributions converge faster.
Pierre-Simon Laplace proved the first general form of the CLT, showing that the sum of many independent errors tends toward a normal distribution.
Jarl Lindeberg and William Feller provided necessary and sufficient conditions for the CLT to hold for independent, non-identically distributed random variables.
Start with n=1: the means histogram mirrors the population. Increase n and watch the bell curve emerge. Notice how the spread shrinks as σ/√n — doubling n shrinks std by 1/√2.
Try Exponential (very skewed) with n=2, 10, 30, 100. Compare how fast Bernoulli vs Chi-Squared converge. With n=1, the means histogram IS the population.