The central limit theorem

In short

• Central limit theorem (CLT): for any population with mean $\mu$ and finite variance $\sigma^2$, the sampling distribution of $\bar X$ from random samples of size $n$ is approximately normal with mean $\mu$ and variance $\sigma^2/n$ when $n$ is large.
• Three properties: approximately normal shape; mean equal to the population mean; variance equal to the population variance divided by $n$.
• Rule of thumb: $n \ge 30$ is large enough. A very non-normal population may need a sample well above 30.
• The population does not have to be normal. The CLT is about the sample mean, not about individual observations.
• Why it matters: it lets us make probability statements about $\mu$, build confidence intervals and test hypotheses even when the population's distribution is unknown or non-normal.