Central Limit Theorem

Precise definition

For independent identically distributed variables $X_1,\ldots,X_n$ with mean $\mu$ and finite positive variance $\sigma^2$, the central limit theorem states $\sqrt n(\bar X-\mu)/\sigma$ converges in distribution to $N(0,1)$ as $n\to\infty$. Thus $\bar X$ is approximately normal with mean $\mu$ and standard error $\sigma/\sqrt n$ for sufficiently large $n$.

Notation and mathematical language

The theorem concerns the sampling distribution of a standardized sum or mean, not the distribution of individual observations. Convergence is asymptotic; there is no universal sample-size cutoff. Skew, heavy tails, dependence, and rare-event probabilities can require much larger $n$.

Conceptual picture

A sum combines many independent contributions. After centring and scaling, no single finite-variance contribution dominates in the iid setting, and the aggregate shape approaches the normal curve. The mean's spread shrinks at rate $1/\sqrt n$.

Fully worked example

Interpretation and application

CLT reasoning supports confidence intervals, quality control, polling, and aggregated risk. It quantifies sampling variability under a model; it does not correct selection bias, measurement bias, confounding, or causal ambiguity.

Common mistakes