Definition
The central limit theorem (CLT) states that the standardised sum of many independent, identically distributed random variables with finite variance approaches a standard normal distribution as the number of terms grows. The sum's mean stays put; its scatter is scaled by the square root of n; what remains is bell-shaped, no matter what the original distribution looked like.
The result is what makes the normal distribution universal: averages and sums of disparate quantities all end up looking the same.
Why it matters
How it works
If X1, X2, ..., Xn are independent and identically distributed with finite mean mu and variance sigma², then the standardised sum (X1 + ... + Xn - n × mu) / (sigma × square root of n) converges in distribution to the standard normal N(0, 1). The approximation is excellent for n above 30 in many practical cases; for skewed distributions, larger n is needed.
In statistics, the CLT explains why the sampling distribution of the mean is approximately normal with mean mu and standard error sigma / square root of n. This single fact powers the bulk of classical inference — confidence intervals, z-tests, and approximate p-values.