What Is the Central Limit Theorem for Sample Means?

Answer

The central limit theorem (CLT) says that, under suitable conditions, the standardized sample mean approaches a standard normal distribution as sample size increases, even when the population itself is not normally distributed.

For independent, identically distributed observations with population mean $\mu$ and finite, positive variance $\sigma^2$,

$$\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\xrightarrow{d}N(0,1).$$

For a sufficiently accurate large-sample approximation, this gives

$$\bar{X}\approx N\!\left(\mu,\frac{\sigma^2}{n}\right),$$

where the second parameter here is variance. The sampling standard deviation, or standard error, is $\sigma/\sqrt{n}$.

What becomes approximately normal?

The distribution of sample means across repeated samples of the same size becomes approximately normal. The theorem does not say that the observations within a large sample, or the underlying population, become normal.

When is the approximation reasonable?

There is no universal rule that $n=30$ always suffices. Strong skewness, heavy tails, or rare extreme values may require much larger samples. The classical statement above also does not justify treating dependent observations as independent or using a finite-variance formula for a population whose variance does not exist.

If the population is normal and observations are independent, the sample mean is exactly normal for every sample size; the CLT is not needed to obtain its shape.