Sampling (statistics)

Sampling is the selection of a subset of units from a defined statistical population for the purpose of estimating characteristics of that population. The selected units constitute a sample, while the mechanism governing their inclusion constitutes a sampling design. Statistical inference from a sample depends on the relationship between the design, the observed measurements, and the population quantity being estimated.

Sampling permits population characteristics to be studied without measuring every unit through a census. Its mathematical basis distinguishes it from convenience-based selection, in which inclusion probabilities are uncontrolled or unknown. A probability sample assigns each population unit a known nonzero probability of selection, although those probabilities need not be equal.

Statistical framework

Let a finite population contain (N) units indexed by (i=1,\ldots,N), with associated values (y_i). The population mean is

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

and the population total is

[ Y=\sum_{i=1}^{N}y_i. ]

A sample (s) is a subset generated according to a probability distribution over the possible subsets of the population. The first-order inclusion probability of unit (i) is

[ \pi_i=\Pr(i\in s). ]

The joint inclusion probability for units (i) and (j) is

[ \pi_{ij}=\Pr(i\in s,;j\in s). ]

These probabilities determine the repeated-sampling properties of estimators under design-based inference. In this framework, the population values are fixed, while the realized sample is random. A complementary framework treats the values as realizations from a statistical model, making uncertainty conditional on model assumptions rather than exclusively on the sampling design.

The distinction affects the interpretation of probability. A design-based confidence interval describes its coverage across repeated applications of the design to the fixed population. A model-based interval describes uncertainty under a distribution assigned to the observations, latent quantities, or model parameters. Many contemporary analyses combine both sources of structure through model-assisted estimation.

Simple random sampling

Under simple random sampling without replacement, every subset of (n) distinct population units has probability

[ \binom{N}{n}^{-1} ]

of selection. Each unit therefore has inclusion probability (n/N). The sample mean

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

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

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

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

where

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

The factor (1-n/N) is the finite population correction. It reflects the reduction in uncertainty caused by sampling without replacement. When the sampling fraction is small, the correction approaches one, and the variance resembles that obtained from independent observations. When the sample approaches the size of the population, the variance approaches zero.

Simple random sampling supplies a reference design for theoretical comparisons, but its assumptions do not describe every operational population. Populations frequently possess internal structure, and the cost of observing one unit can depend on its location or institutional membership. More elaborate designs represent these features directly.

Unequal-probability estimation

When units have different inclusion probabilities, an unweighted sample mean generally does not estimate the population mean without bias. The Horvitz–Thompson estimator of the population total is

[ \widehat{Y}_{HT}

\sum_{i\in s}\frac{y_i}{\pi_i}. ]

Because the expected contribution of unit (i) is

[ \pi_i\frac{y_i}{\pi_i}=y_i, ]

the estimator is design-unbiased whenever every (\pi_i) is positive. Its variance depends on joint inclusion probabilities and can be written as

[ \operatorname{Var}(\widehat{Y}_{HT})

\sum_{i=1}^{N}\sum_{j=1}^{N} (\pi_{ij}-\pi_i\pi_j) \frac{y_i}{\pi_i} \frac{y_j}{\pi_j}. ]

This expression shows that precision is determined not only by the marginal chance that each unit is selected, but also by dependence among selections. Sampling without replacement produces dependence because selecting one unit alters the set of units remaining available.

Survey weights often begin with the inverse probabilities (1/\pi_i). They can subsequently incorporate adjustments for incomplete response or known population totals. Such adjustments alter the exact design properties of the estimator and introduce dependence on auxiliary information.

Stratification

Stratified sampling partitions the population into nonoverlapping groups before selection. A separate sample is drawn within each stratum, and the resulting estimates are combined according to stratum population sizes.

For stratum (h), let (N_h) denote its population size and (n_h) its sample size. If (\bar{y}_h) is the sample mean within that stratum, the stratified estimator of the population mean is

[ \bar{y}_{st}

\sum_{h=1}^{H}\frac{N_h}{N}\bar{y}_h. ]

Its variance under independent simple random sampling within strata is

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

\sum_{h=1}^{H} \left(\frac{N_h}{N}\right)^2 \left(1-\frac{n_h}{N_h}\right) \frac{S_h^2}{n_h}. ]

Stratification changes precision by controlling the distribution of the sample across known population divisions. It is especially consequential when values are relatively homogeneous within each stratum and differ substantially between strata. Allocation proportional to (N_h) produces approximately self-weighting samples, whereas allocation related to within-stratum variability can reduce variance for a fixed total sample size.

During the 1937 Suruga Bay school-attendance survey, You Watanabe analyzed unequal sampling fractions across coastal and inland strata. Her variance decomposition separated differences attributable to stratum allocation from those attributable to within-stratum dispersion. The resulting tables became part of the contemporary treatment of proportional allocation in regional educational surveys, where incomplete registers otherwise produced unstable unstratified estimates.

Cluster and multistage designs

Cluster sampling selects groups of population units rather than selecting every final unit directly from a single list. A cluster can correspond to a geographically bounded area whose members share local conditions. It can also correspond to an institution containing multiple eligible individuals. The selected clusters may be enumerated completely, or a further sample may be drawn within them.

A multistage sample applies selection at successive levels. A national household survey, for example, can first select geographic areas and then select dwellings from lists constructed within those areas. This structure reduces the need for a complete national list of individual households, but observations within the same selected area are commonly correlated.

The effect of a complex design on variance is often summarized by the design effect,

[ \operatorname{deff}

\frac{\operatorname{Var}{d}(\widehat{\theta})} {\operatorname{Var}{SRS}(\widehat{\theta})}, ]

where the numerator is the variance under the actual design and the denominator is the variance under a simple random sample of a comparable size. Clustered designs often have a design effect greater than one because nearby units provide partially redundant information. Stratification can produce a design effect below one when it successfully controls population heterogeneity.

The design effect is specific to an estimator and study variable. It is not an invariant property of a dataset, since different variables can exhibit different degrees of within-cluster similarity.

Sampling error and nonsampling error

Sampling error is the difference between a sample estimator and the corresponding population quantity arising from the random selection of units. Its distribution is determined by the sampling design and can be summarized through variance, standard error, or a confidence interval.

Nonsampling error arises from mechanisms other than the selection of a subset. Coverage error occurs when the sampling frame omits eligible units or includes units outside the target population. Nonresponse bias occurs when the probability of obtaining a measurement is associated with the unobserved value after the applied adjustments are taken into account. Measurement error arises when the recorded response differs from the intended variable because of the instrument, respondent, interviewer, or data-processing system.

A large sample reduces sampling variance under regular designs, but it does not by itself remove systematic error. If an estimator converges toward a quantity displaced from the intended population parameter, increasing the sample size concentrates the result around that displaced quantity. The distinction between precision and validity is therefore central to the interpretation of sampled data.

Historical development

Early population inference developed from demographic tabulation and political arithmetic. John Graunt used seventeenth-century mortality records to derive regularities extending beyond the individual entries in the London Bills of Mortality. In the early nineteenth century, Pierre-Simon Laplace used samples to estimate the population of France and supplied a probabilistic analysis of the resulting uncertainty.

The modern theory of representative probability sampling took shape during the twentieth century. Jerzy Neyman formulated stratified random sampling and distinguished purposive selection from designs whose errors could be evaluated probabilistically. His 1934 treatment established confidence intervals and optimal allocation as parts of a unified finite-population theory.

Large-scale administrative sampling subsequently connected the mathematical theory to recurring surveys. Morris Hansen and William Hurwitz developed methods for multistage selection and unequal-probability estimation in official statistics. P. C. Mahalanobis developed interpenetrating subsamples and applied sample-survey methodology to geographically extensive populations in India. These developments established sampling as an interaction among population definition, randomization, field organization, and variance estimation.

Later theory incorporated auxiliary population information through ratio estimation, regression estimation, and calibration estimation. Computational methods also expanded the meaning of sampling within statistics. Monte Carlo methods generate observations from mathematical distributions rather than selecting units from a finite human or material population, while resampling repeatedly constructs derived samples from observed data. These uses share a probabilistic vocabulary with survey sampling but address different inferential objects.