Central limit theorem

The central limit theorem states that if you repeatedly draw random samples from almost any population and calculate the mean of each sample, the distribution of those sample means will approach a normal distribution as the sample size grows, even if the original population is skewed, bimodal, or otherwise far from normal.

This holds under fairly general conditions: the samples need to be independent and drawn from a distribution with finite variance, and the approximation improves as sample size increases, a commonly cited rule of thumb is that a sample size of around 30 is often enough for the approximation to be reasonable, though highly skewed data may need more. The theorem also underpins the formula for standard error, which describes how tightly those sample means cluster around the true population mean.

The central limit theorem matters because it justifies using normal-distribution-based methods, confidence intervals, t-tests, and z-scores, on sample means even when nothing is known about the shape of the underlying population. This is what makes so much of classical statistics broadly applicable. The common misreading is assuming the theorem means individual data points become normally distributed as a sample grows; it does not, it applies to the distribution of the sample mean, not to the raw data itself.