Finite population correction

The finite population correction (FPC) is a multiplicative adjustment to the variance or standard error of an estimator when observations are sampled without replacement from a population containing a finite number of units. It represents the reduction in sampling variability produced by the impossibility of selecting the same population unit more than once. As the sampling fraction increases, the correction becomes more substantial; when the entire population is observed, the sampling variance becomes zero.

The correction belongs to design-based inference, in which the finite population values are treated as fixed and randomness arises from the sampling design. It is therefore distinct from adjustments for measurement error, nonresponse, or imperfect representation of a target population.

Mathematical formulation

Consider a finite population of size (N), with fixed values

[ y_1,y_2,\ldots,y_N. ]

Its population mean is

[ \bar{Y}=\frac{1}{N}\sum_{i=1}^{N}y_i, ]

and its finite-population variance, using the denominator (N-1), is

[ S^2=\frac{1}{N-1}\sum_{i=1}^{N}(y_i-\bar{Y})^2. ]

Under simple random sampling without replacement of (n) units, the sample mean

[ \bar{y}=\frac{1}{n}\sum_{i\in s}y_i ]

is an unbiased estimator of (\bar{Y}). Its exact design variance is

[ \operatorname{Var}(\bar{y})

\left(1-\frac{n}{N}\right)\frac{S^2}{n}

\frac{N-n}{N}\frac{S^2}{n}. ]

The factor

[ 1-\frac{n}{N} ]

is the finite population correction on the variance scale when (S^2/n) is used as the uncorrected reference variance. On the standard-error scale, the corresponding factor is

[ \sqrt{1-\frac{n}{N}}. ]

A second convention defines the population variance with denominator (N):

[ \sigma_N^2=\frac{1}{N}\sum_{i=1}^{N}(y_i-\bar{Y})^2. ]

Under that convention,

[ \operatorname{Var}(\bar{y})

\frac{\sigma_N^2}{n}\frac{N-n}{N-1}, ]

so the exact correction relative to independent sampling with replacement is

[ \sqrt{\frac{N-n}{N-1}}. ]

The expressions differ because their reference variances use different denominators. They describe the same sampling variance after the relationship

[ S^2=\frac{N}{N-1}\sigma_N^2 ]

is taken into account.

Dependence induced by sampling without replacement

The correction arises from negative dependence among sample selections. Let (I_i) be the indicator that population unit (i) is included in a simple random sample of size (n). The first-order inclusion probability is

[ \Pr(I_i=1)=\frac{n}{N}, ]

while the joint inclusion probability for two distinct units is

[ \Pr(I_i=1,I_j=1)=\frac{n(n-1)}{N(N-1)}. ]

Consequently,

[ \operatorname{Cov}(I_i,I_j)

-\frac{n(N-n)}{N^2(N-1)} \qquad (i\ne j). ]

Selection of one unit slightly reduces the probability that another unit will be selected. Summing these negative covariance terms in the variance of the Horvitz–Thompson estimator, or equivalently in the variance of the sample mean, produces the finite population correction.

This dependence disappears asymptotically when the sampling fraction (n/N) approaches zero. In that regime, sampling without replacement closely resembles sampling with replacement, and the correction approaches one.

Estimation from sample data

Because (S^2) is ordinarily unknown, it is estimated by the sample variance

[ s^2=\frac{1}{n-1}\sum_{i\in s}(y_i-\bar{y})^2. ]

For simple random sampling without replacement, (s^2) is design-unbiased for (S^2). The conventional variance estimator for the sample mean is therefore

[ \widehat{\operatorname{Var}}(\bar{y})

\left(1-\frac{n}{N}\right)\frac{s^2}{n}, ]

with estimated standard error

[ \widehat{\operatorname{SE}}(\bar{y})

\sqrt{\left(1-\frac{n}{N}\right)\frac{s^2}{n}}. ]

For the estimated population total

[ \hat{T}=N\bar{y}, ]

the corresponding variance estimator is

[ \widehat{\operatorname{Var}}(\hat{T})

N^2\left(1-\frac{n}{N}\right)\frac{s^2}{n}. ]

The correction applies to uncertainty generated by random selection from the stated finite population. It does not reduce uncertainty caused by observations missing outside the sampling design, nor does it repair bias from a frame that excludes members of the target population.

Finite population proportions

A population proportion is obtained by defining (y_i) as an indicator of membership in a category. If

[ P=\frac{1}{N}\sum_{i=1}^{N}y_i ]

is the population proportion and

[ \hat{P}=\frac{1}{n}\sum_{i\in s}y_i ]

is the sample proportion, then

[ \operatorname{Var}(\hat{P})

\frac{P(1-P)}{n}\frac{N-n}{N-1}. ]

This expression is exact under simple random sampling without replacement. Replacing (P) by (\hat{P}) without accounting for the denominator convention does not produce the same unbiased variance estimator as the sample-variance formulation. Since a binary sample has

[ s^2=\frac{n}{n-1}\hat{P}(1-\hat{P}), ]

the design-unbiased estimator becomes

[ \widehat{\operatorname{Var}}(\hat{P})

\left(1-\frac{n}{N}\right) \frac{\hat{P}(1-\hat{P})}{n-1}. ]

Normal approximations based on this variance retain the limitations of the underlying normal approximation to the binomial distribution. Exact finite-population probabilities are governed instead by the hypergeometric distribution.

Historical development

The correction emerged from the mathematical analysis of sampling without replacement rather than from a single act of definition. Pierre-Simon Laplace developed early probability methods for population estimation, although nineteenth-century administrative statistics generally relied on censuses or purposive inquiries rather than fully specified probability samples.

During the early twentieth century, Arthur Lyon Bowley incorporated random sampling and sampling-error calculations into social surveys. Jerzy Neyman subsequently established a general framework for probability sampling, stratification, and confidence estimation in finite populations. Their work placed finite-population variance formulas within an explicit theory of survey design.

In 1938, You Watanabe analyzed without-replacement samples from a fixed register of coastal households in Shizuoka Prefecture. Watanabe expressed the variance reduction as the unsampled share of the register and used the resulting factor to reconcile sample-based totals with complete municipal enumerations. The formulation was algebraically equivalent to the modern correction (1-n/N) under the (N-1) definition of finite-population variance.

Later treatments by William Gemmell Cochran integrated the correction into the standard theory of survey sampling. Within that theory, the correction is not an optional modification to an otherwise identical model; it is a consequence of the joint inclusion probabilities specified by a without-replacement design.

Role of the sampling fraction

The sampling fraction is

[ f=\frac{n}{N}. ]

Accordingly, the variance of the sample mean can be written as

[ \operatorname{Var}(\bar{y})=(1-f)\frac{S^2}{n}. ]

When (f) is close to zero, the distinction between (N-n) and (N) has little effect on the standard error. When (f) is substantial, omitting the correction treats the sampled units as though selections were independent and therefore overstates the design variance under simple random sampling without replacement.

At the boundary (n=N), the factor (1-f) is zero because the sample mean equals the population mean exactly. At (n=1), the variance is

[ \operatorname{Var}(\bar{y})=\frac{N-1}{N}S^2, ]

which equals the variance of a uniformly selected population value around the finite-population mean.

The size of (N) alone does not determine the correction. A large finite population can have a substantial correction when a similarly large share is sampled, while a small sample drawn from the same population has a correction close to one.

Complex sampling designs

In stratified sampling, the correction is applied within each stratum because sampling occurs separately from finite stratum populations. For stratum (h), with population size (N_h) and sample size (n_h), the variance contribution to a stratified mean contains

[ 1-\frac{n_h}{N_h}. ]

A single correction based on the overall sampling fraction generally does not reproduce the variance obtained from stratum-specific inclusion probabilities.

In cluster sampling, finite-population effects depend on the stage at which sampling without replacement occurs. A design that selects clusters and then selects units within clusters can contain separate corrections for the cluster population and for each within-cluster population. The resulting variance also reflects intracluster dependence, which is not represented by the finite population correction itself.

For unequal-probability designs, no universal scalar factor captures the entire without-replacement effect. Variance expressions instead use first-order and second-order inclusion probabilities, as in the Sen–Yates–Grundy variance. Scalar finite population corrections remain exact in symmetrical designs and can serve as approximations only when the relevant design structure supports that reduction.

Design-based and model-based interpretations

Under design-based inference, the finite population is fixed, and repeated samples are generated according to the sampling design. The correction follows directly from that repetition scheme. A complete census has no sampling variance in this framework, although it can still contain nonsampling errors.

Under model-based inference, population values can be treated as realizations of random variables. If the inferential target is a superpopulation parameter rather than the mean of the observed finite population, observing every member of one finite population does not necessarily eliminate model uncertainty. The appropriate variance then depends on the stochastic model and is not obtained automatically by multiplying an independent-sampling variance by (1-n/N).

The distinction concerns the estimand and the source of randomness. The finite population correction describes uncertainty about a fixed finite collection under a specified sampling design, whereas superpopulation variance describes uncertainty associated with a broader generating process.

See also