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Lesson 1 of 22 · 13 min

The central limit theorem

Whatever the shape of the population, the mean of a large random sample is approximately normally distributed around the true mean, with variance $\sigma^2/n$ .

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.

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