The Design Expectation of the Finite-Population Sample Variance

Why the Sampling Design Matters

A sample variance is calculated from the observations selected, so it changes from sample to sample. To ask whether it estimates a population variance without bias, we must specify what is random: in this tutorial, the population values are fixed and the randomness comes entirely from the sampling design. Under simple random sampling without replacement, every subset of the population of the specified size has the same chance of selection.

With the usual finite-population convention, both the population variance and the sample variance use a degrees-of-freedom denominator: \(N-1\) for the population and \(n-1\) for the sample. The central result is that the sample variance is unbiased for the finite-population variance under this design. The proof rests on expressing each variance as an average of squared pairwise differences.

Population and Sample Variances

The restriction \(n\geq 2\) is necessary because the usual sample variance has denominator \(n-1\). When \(n=N\), the sample is the entire population, so \(s_s^2=S_U^2\) for every possible selection (there is only one).

A Pairwise-Difference Identity

The key algebraic fact is that centered sums of squares can be written using all pairs of values. It applies to any fixed list, whether that list is a sample or the whole population.

Proof. Expand each squared difference. Each \(x_i^2\) appears in exactly \(m-1\) pairs, and the cross-products together give \(-2\sum_{i<j}x_ix_j\). Thus

The second equality uses \((\sum_i x_i)^2=\sum_i x_i^2+2\sum_{i<j}x_ix_j\). For the last equality, expanding the centered sum gives \(\sum_i x_i^2-2\bar x\sum_i x_i+m\bar x^2=\sum_i x_i^2-m\bar x^2\). This proves the identity. \(\square\)

Dividing the identity by the appropriate factors gives two useful representations:

For example, in the sample representation, the lemma says that the pairwise sum is \(n\sum_{j=1}^n(Y_{i_j}-\bar Y_s)^2\). Dividing by \(n(n-1)\) therefore gives exactly the usual sample variance. The population representation follows by applying the same identity to all \(N\) values.

The Design-Unbiasedness Theorem

Proof. For each population pair \(i<j\), let \(I_{ij}\) equal 1 if both units \(i\) and \(j\) are included in the sample and 0 otherwise. The sample pairwise representation can be written as

All population values are fixed. Under SRSWOR, the probability that a particular pair is included is the fraction of size-\(n\) subsets containing both units:

The last equality follows by expanding the binomial coefficients. Taking expectation of the finite sum and substituting this same inclusion probability for every pair gives

The final equality is the population pairwise-difference representation. This proves the theorem. \(\square\)

The argument is finite: expectation is interchanged with a finite sum, so no limit or integrability condition is needed. The essential design property is the common probability that any particular pair is included.

Worked Calculations

What the Denominator Means

Some disciplines define a finite-population variance with denominator \(N\), rather than \(N-1\). If that convention is used, write

Because \(S_U^2=\frac{N}{N-1}V_U\), the theorem implies \(E_d[s_s^2]=\frac{N}{N-1}V_U\), not \(V_U\). The usual sample variance is unbiased for the \(N-1\) convention. Stating the denominator is therefore part of stating the estimand; the word “variance” alone does not settle which finite-population quantity is intended.

A related derived result describes the numerator of the sample variance. Since \(s_s^2\) is the sample centered sum of squares divided by \(n-1\), multiplying the theorem by \(n-1\) gives

The result concerns a fixed finite population and a random sample design. It does not require a probability model for how the population values were generated.

Common Pitfall

• Wrong reasoning: “The sample variance divides by \(n-1\), so it must be unbiased for any population variance, regardless of its definition or sampling method.”
• Correction: The target here is specifically \(S_U^2\), with denominator \(N-1\), and the proof uses the equal pair-inclusion probability under SRSWOR. If the target instead uses denominator \(N\), the expectation differs by the factor \(N/(N-1)\). If the selection design changes, the pair-inclusion probabilities must be reconsidered rather than assumed to give the same result.

The practical lesson is to name both the target and the design before claiming unbiasedness. For SRSWOR, writing the sample variance as a pairwise sum exposes exactly why its design expectation matches the \(N-1\) finite-population variance.