6.4.3—Sample mean

For independent observations Xᵢ with mean μ and variance σ², the sample mean X̄ has E(X̄)=μ and Var(X̄)=σ²/n. Its standard error is σ/√n (or estimated with s/√n).

Distinguish variability of individual observations from variability of means, and account for finite-population or dependence conditions when relevant.

If σ=12 and n=36, the standard error of X̄ is 2, even though individual values vary with standard deviation 12.

Increasing n reduces standard error by √n, not by n, and does not necessarily reduce measurement bias.