Central limit theorem

Let $X_1, \dots, X_n$ be independent draws from one population (independent and identically distributed) with mean $\mu$ and variance $0 < \sigma^2 < \infty$, and let $\bar X$ be their average. The central limit theorem, in its Lindeberg–Lévy form, says that the standardized average converges in distribution to the standard normal: its distribution function tends to $\Phi$ at every point. Informally, $\bar X \approx N(\mu, \sigma^2/n)$. The page shows how the shape of the distribution of $\bar X$ changes with n.

The upper layer is the population. The lower layer shows the distribution of the average $\bar X$ of $n$ draws. Repeating this $N$ times gives the yellow histogram: it fluctuates with sampling and can be compared with the theoretical curve. A fixed seed reproduces the same samples. Magnified by √n, the lower axis is $Y = \mu + \sqrt{n}\,(\bar x - \mu)$: its mean μ and standard deviation σ match the population, making the shapes easier to compare.

Curves marked “numeric” are computed numerically; the others use exact distribution formulas. Skewness measures asymmetry. Excess kurtosis compares the standardized fourth moment with that of a normal distribution, for which it is zero. When the corresponding population moments exist, the average has skewness $\gamma/\sqrt{n}$ and excess kurtosis $\kappa/n$. Neither measure alone guarantees a close normal approximation.