AHL 4.15 (HL)—Sampling distributions and CLT

A sampling distribution is the probability distribution of a statistic, such as a sample mean, over repeated samples of the same size. It is not the distribution of individual observations.

For independent observations with population mean μ and standard deviation σ, the sample mean has mean μ and standard deviation σ/√n. The central limit theorem makes the shape approximately normal when n is sufficiently large under suitable conditions.

If μ=50, σ=12 and n=36, the standard error of the mean is 2. A sample mean of 54 is two standard errors above μ; that statement concerns sampling variation, not one student's score.

Increasing n reduces the standard error, not the population spread σ. The CLT does not erase dependence, extreme bias or a badly defined sampling process.

A linear combination of independent normal variables is exactly normal. If the population is normal, $\bar X\sim N(\mu,\sigma^2/n)$ for any $n$; for a general population the central limit theorem makes $\bar X$ approximately normal as $n$ grows, with $n>30$ treated as sufficient in examinations. Use variance $\sigma^2/n$ or standard error $\sigma/\sqrt n$ consistently.