What is the distribution of the sample mean for large n?

• A. Approximately Normal with mean μ and variance σ^2 / n
• B. Uniform distribution on [0,1]
• C. Exponential distribution with rate λ
• D. Binomial distribution with parameters n and p

Answer: (A)

Central limit theorem tells us that if you take n independent observations with population mean μ and variance σ^2, the distribution of their average is approximately Normal with mean μ and variance σ^2/n. In other words, the standard deviation is σ/√n, so as n grows the spread around μ tightens. If the underlying population is already normal, the sample mean is exactly Normal with those parameters for any n; otherwise, for large n the normal approximation is very good. The other distributions (uniform, exponential, binomial) describe different kinds of randomness and do not generally describe the sampling distribution of a mean.