Central limit theorem

The central limit theorem (CLT) says that if you take sufficiently large random samples from any population — no matter how skewed or oddly shaped — the distribution of the sample means will be approximately normal. The approximation improves as sample size n increases, and a common rule of thumb is that n ≥ 30 is large enough for most populations.

The theorem also pins down the sampling distribution's center and spread. The mean of the sampling distribution equals the population mean μ, and its standard deviation — called the standard error — equals σ/√n, the population standard deviation divided by the square root of the sample size. Larger samples therefore produce sample means that cluster more tightly around μ.

A concrete example: household incomes are strongly right-skewed, so a single income is far from normally distributed. But if you repeatedly draw samples of 50 households and record each sample's mean income, those means pile up in a nearly perfect bell curve centered on the true population mean. This is what lets statisticians use normal-based methods — z-scores, confidence intervals, significance tests — on data from non-normal populations.

The CLT is a cornerstone topic on the AP Statistics exam, which tests when the normal approximation applies and how standard error shrinks with sample size, and it appears on the FE Mechanical exam within its probability and statistics coverage of common distributions. For either exam, remember: the population can be any shape; it's the distribution of sample means that becomes normal.