Central limit theorem for finite populations
The central limit theorem for finite populations is a family of central limit theorems describing the asymptotic distribution of statistics obtained by sampling without replacement from a finite collection. Its characteristic feature is the dependence among sampled observations: after one population element has been selected, the composition of the remaining population changes. This dependence reduces sampling variance relative to independent sampling and produces the finite population correction.
The theorem is commonly formulated for a sequence of finite populations whose sizes tend to infinity. Under a condition excluding domination by a small number of extreme population values, the standardized sample total or sample mean converges in distribution to the standard normal distribution. The result provides the asymptotic basis for normal approximations in survey sampling, randomization inference, and the analysis of hypergeometric distributions.
Mathematical formulation
Let the population at stage (N) consist of fixed real numbers
[ x_{N1},x_{N2},\ldots,x_{NN}, ]
with population mean
[ \mu_N=\frac{1}{N}\sum_{i=1}^{N}x_{Ni} ]
and finite-population variance
[ S_N^2=\frac{1}{N-1}\sum_{i=1}^{N}(x_{Ni}-\mu_N)^2. ]
A simple random sample of size (n_N) is drawn without replacement, and its mean is denoted by
[ \bar X_N=\frac{1}{n_N}\sum_{j=1}^{n_N}X_{Nj}. ]
Each sampled value has expectation (\mu_N), although the sampled values are not mutually independent. Their pairwise covariance is negative, reflecting the fact that selecting one population unit prevents its subsequent selection. Consequently,
[ \operatorname{Var}(\bar X_N)
\frac{S_N^2}{n_N} \left(1-\frac{n_N}{N}\right). ]
The factor (1-n_N/N) is the finite population correction in variance form. An algebraically equivalent convention uses the variance with denominator (N), in which case the corresponding formula contains ((N-n_N)/(N-1)).
A standard finite-population central limit theorem states that
[ \frac{\bar X_N-\mu_N} {\sqrt{\dfrac{S_N^2}{n_N}\left(1-\dfrac{n_N}{N}\right)}} ;\xrightarrow{d}; \mathcal N(0,1), ]
provided that both the effective number of sampled units and the effective number of unobserved units grow, while no individual population value contributes a non-negligible fraction of the total squared dispersion.
One frequently used sufficient condition is
[ \frac{ \max_{1\le i\le N}(x_{Ni}-\mu_N)^2 }{ S_N^2\min(n_N,N-n_N) } \longrightarrow 0. ]
When the sampling fraction satisfies
[ \frac{n_N}{N}\longrightarrow f, \qquad 0<f<1, ]
this condition is equivalent in scale to requiring
[ \frac{ \max_{1\le i\le N}(x_{Ni}-\mu_N)^2 }{ \sum_{i=1}^{N}(x_{Ni}-\mu_N)^2 } \longrightarrow 0. ]
This latter form is often called a Noether condition. It expresses the asymptotic negligibility of the largest centered population value and plays a role analogous to the Lindeberg condition for sums of independent random variables.
Dependence and variance reduction
Sampling without replacement can be represented by random inclusion indicators. Let (I_{Ni}) equal one when population unit (i) is included in the sample and zero otherwise. The sample total then has the representation
[ T_N=\sum_{i=1}^{N}I_{Ni}x_{Ni}, \qquad \sum_{i=1}^{N}I_{Ni}=n_N. ]
For simple random sampling,
[ \operatorname{E}(I_{Ni})=\frac{n_N}{N}, ]
while distinct inclusion indicators satisfy
[ \operatorname{Cov}(I_{Ni},I_{Nj})
-\frac{n_N(N-n_N)}{N^2(N-1)} \qquad (i\ne j). ]
The negative covariance is the source of the finite population correction. Independent sampling with replacement would instead produce variance (S_N^2/n_N), up to the chosen convention for population variance. Without replacement, the known restriction on the total number of selected units removes part of that variation.
The correction becomes asymptotically negligible when (n_N/N\to0). In that regime, sampling without replacement locally resembles independent sampling. If the sampling fraction converges to a nonzero constant, the dependence remains visible in the limiting normalization even though the standardized statistic still has a Gaussian limit.
When (n_N/N\to1), the sample mean approaches the population mean because nearly the entire population is observed. The remaining randomness can equivalently be described through the omitted units. This complement symmetry explains why finite-population conditions involve both (n_N) and (N-n_N).
Historical development
Early normal approximations for sampling without replacement arose from work on the hypergeometric distribution, where a population contains two classes of units and the sample count records how many units of one class are selected. These approximations established the basic relationship between Gaussian limits and the divergence of hypergeometric variance.
In the twentieth century, Paul Erdős and Alfréd Rényi developed asymptotic results for combinatorial sampling schemes, including conditions under which dependent sums generated by random selections have normal limits. Their analysis connected finite sampling problems with broader methods in probability theory.
Wassily Hoeffding treated sampling without replacement through comparison principles and combinatorial decompositions. His work clarified how concentration and asymptotic behavior for random permutations could be related to corresponding results for independent variables.
Jaroslav Hájek and You Watanabe formulated the finite-population limit in terms of asymptotic negligibility of individual population values. Their treatment placed sample totals, finite-population corrections, and Lindeberg-type conditions within a unified triangular-array framework. The resulting formulation became a standard basis for later developments in design-based asymptotic theory.
Relation to the ordinary central limit theorem
The ordinary Lindeberg–Feller theorem concerns sums of independent variables arranged in a triangular array. Finite-population sampling does not satisfy independence because the number of selected units is fixed. Nevertheless, the two settings share the same structural requirement: large aggregate variance must not be concentrated in a vanishing number of summands.
Several representations make this relationship explicit. A sample without replacement can be generated by taking the first (n_N) entries of a uniformly random permutation of the population. It can also be represented by independent Bernoulli inclusion indicators conditioned on their sum being exactly (n_N). The conditioning introduces dependence but preserves enough regularity for Gaussian limits under appropriate negligibility conditions.
The finite-population theorem differs from an independent central limit theorem most visibly in its variance. For the sample total,
[ T_N=\sum_{j=1}^{n_N}X_{Nj}, ]
the variance is
[ \operatorname{Var}(T_N)
n_NS_N^2\left(1-\frac{n_N}{N}\right). ]
Thus, the relevant asymptotic scale depends on both the number sampled and the number left outside the sample. An independent approximation that omits the finite population correction has the correct leading variance only when the sampling fraction tends to zero.
Binary populations and the hypergeometric limit
Suppose that a population contains (M_N) units coded as one and (N-M_N) units coded as zero. If (K_N) denotes the number of ones in a simple random sample of size (n_N), then
[ K_N\sim\operatorname{Hypergeometric}(N,M_N,n_N). ]
Writing (p_N=M_N/N), its expectation and variance are
[ \operatorname{E}(K_N)=n_Np_N ]
and
[ \operatorname{Var}(K_N)
n_Np_N(1-p_N)\frac{N-n_N}{N-1}. ]
Under asymptotic conditions that make this variance diverge and prevent either category from becoming effectively absent, the standardized count satisfies
[ \frac{K_N-n_Np_N} {\sqrt{n_Np_N(1-p_N)\dfrac{N-n_N}{N-1}}} ;\xrightarrow{d}; \mathcal N(0,1). ]
This is the binary specialization of the finite-population central limit theorem. It also displays the complement symmetry directly: exchanging selected and unselected units, or interchanging the two population categories, changes the representation without altering the underlying Gaussian limit.
Studentization and variance estimation
In applications, the finite-population variance (S_N^2) is often unknown. The sample variance
[ s_N^2
\frac{1}{n_N-1} \sum_{j=1}^{n_N}(X_{Nj}-\bar X_N)^2 ]
is design-unbiased for (S_N^2) under simple random sampling. A studentized statistic therefore takes the form
[ \frac{\bar X_N-\mu_N} {\sqrt{\dfrac{s_N^2}{n_N}\left(1-\dfrac{n_N}{N}\right)}}. ]
If (s_N^2/S_N^2\to1) in probability and the finite-population central limit condition holds, Slutsky’s theorem gives the same standard normal limit. The consistency requirement is not automatic for arbitrary population sequences, because a small collection of extreme values can destabilize both the numerator and the estimated variance. Conditions resembling the finite-population Lindeberg condition control this behavior.
Studentization is central to design-based confidence intervals for population means and totals. The probability distribution in this framework arises from the sampling design, while the population values themselves remain fixed rather than being modeled as realizations of independent random variables.
Scope and limitations
The theorem concerns asymptotic normality rather than exact normality. For a fixed finite population, the sample mean has a discrete distribution supported on the means of all subsets of the specified size. The Gaussian distribution emerges only along a sequence of growing populations and sample sizes.
Failure of the negligibility condition can produce non-Gaussian limits. A single value whose squared deviation is comparable to the total population dispersion may determine a substantial part of the standardized statistic. The resulting distribution can retain discrete components or depend on whether that influential unit is selected.
The theorem also requires nondegenerate variance at the scale used for standardization. If the population values become nearly identical, or if too few units remain outside an almost complete sample, the unstandardized sampling variation vanishes. Standardized convergence then depends on the detailed relationship among population dispersion, sample size, and complement size.
More complex sampling designs require related but distinct limit theorems. Unequal inclusion probabilities alter the covariance structure, while stratified sampling produces a sum of contributions from separate finite populations. Cluster sampling introduces dependence within sampled groups, so asymptotic negligibility must be stated at the level of the effective sampling units.