The central limit theorem

Central limit theorem (CLT). Take independent observations, all from the same population, with mean $\mu$ and finite standard deviation $\sigma$; the population can have any shape. As the sample size $n$ grows, the sampling distribution of the sample mean $\bar{x}$ gets closer and closer to a normal distribution with mean $\mu$ and standard deviation $\sigma/\sqrt{n}$.

Three things the CLT does not say: that the data become normal (a sample of 1 000 skewed commutes is still skewed); that $n = 30$ is always enough; or that it applies to every statistic in the same way (it is a statement about means and sums, and proportions as means of 0/1 values).

In practice “independent” means a random sample with replacement, or a random sample without replacement that is a small fraction of the population (under about 10%). When a larger fraction is taken, the means are still close to normal but their SD is smaller than $\sigma/\sqrt{n}$: that is the finite-population correction, later in this track.