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Finite-population variance of a simple random mean

For a size-n simple random sample without replacement from N fixed units, the mean variance is (1 − n/N)S²/n.

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Fix the variance convention

S2=∑i=1N(xi−μ)2N−1,Var⁡(Xˉ)=(1−nN)S2nS^2=\frac{\sum_{i=1}^{N}(x_i-\mu)^2}{N-1},\quad \operatorname{Var}(\bar X)=\left(1-\frac nN\right)\frac{S^2}{n}

N > 1 and 1 ≤ n ≤ N; the randomness comes from uniform selection of size-n subsets of the fixed finite population.

Check the full-population limit

When n = N, the sample contains all units and its mean equals the finite population mean on every draw. The selection variance is zero. Measurement errors or missing outcomes can still create problems, but they are not this subset-selection variance.

Compare the with-replacement convention

For independent uniform draws with replacement, the finite value-distribution variance uses divisor N. Call it sigma_N² = [(N − 1)/N]S². The mean variance for n independent draws is sigma_N²/n. Mixing sigma_N² with the displayed S² formula loses the divisor adjustment.

Require the design that the formula describes

A systematic sample with equal unit chances does not automatically have this variance. Neither does a clustered sample or a frame altered by nonresponse. Read the selection mechanism before applying a finite-population correction.

Further references