6.4.5—Sample mean

For independent observations with mean μ and variance σ², E(X̄)=μ and Var(X̄)=σ²/n. A linear transformation aX̄+b has mean aμ+b and variance a²σ²/n.

Separate the spread of individual observations from the standard error of the mean, and state independence assumptions before using the formula.

If σ=10 and n=25, the mean has standard deviation 2; quadrupling n halves the standard error.

A sample mean is not guaranteed to equal μ, and multiplying by a changes variance by a², not a.