Central Limit Theorem

The central limit theorem gives conditions under which centered and scaled sums or means approach a normal distribution as sample size grows.

In its familiar independent, identically distributed form, the central limit theorem says that a properly centered and scaled sum approaches a normal distribution when the variables have a finite mean and finite, nonzero variance. Equivalently, the standardized sample mean approaches normality. The unscaled mean concentrates near the population mean rather than approaching a fixed bell curve of unchanged width.

The theorem is asymptotic: it does not supply one sample size that makes every normal approximation accurate. Strong skew, rare outcomes, dependence, and infinite variance can invalidate a casual application or require other results. It also does not turn a biased sample into a representative one. Check the data-generating process and the accuracy needed for the actual inference, especially in the tails.

When to use it

When reasoning about the sampling distribution of a sum or mean under a specified data-generating process.

How it can help

Check sampling, dependence, moments, and finite-sample accuracy before using a normal approximation for an average.