Simple random sample

A simple random sample is a probability sample in which every subset of a specified size has the same probability of selection from a finite population. The term usually denotes sampling without replacement, so a population unit cannot occur more than once in the resulting sample. This equal-probability property distinguishes simple random sampling from designs that deliberately assign different selection probabilities or impose restrictions based on auxiliary information.

Simple random sampling provides a fundamental reference design in survey sampling and design-based inference. Its mathematical structure permits exact calculation of inclusion probabilities, estimator variances, and sampling distributions. More elaborate designs are frequently analyzed by comparing their statistical behavior with that of a simple random sample of the same size.

Definition

Let the finite population be

[ U={1,2,\ldots,N}, ]

and let (s\subseteq U) denote a sample containing (n) distinct units. Under simple random sampling without replacement, every possible sample of size (n) has probability

[ P(S=s)=\binom{N}{n}^{-1}. ]

The design therefore has exactly (\binom{N}{n}) possible unordered outcomes. If selection order is retained, each ordered sequence of (n) distinct units has probability

[ \frac{1}{N(N-1)\cdots(N-n+1)}. ]

Equal probabilities for complete samples imply equal first-order inclusion probabilities. For every population unit (i),

[ \pi_i=P(i\in S)=\frac{n}{N}. ]

For two distinct units (i) and (j), the joint inclusion probability is

[ \pi_{ij}=P(i,j\in S)=\frac{n(n-1)}{N(N-1)}. ]

The corresponding inclusion indicators are negatively correlated because selecting one unit leaves fewer available positions for the remaining units. This dependence is the source of the finite population correction.

Sampling with replacement is related but mathematically distinct. In that design, each draw independently selects one of the (N) population units with probability (1/N), and a unit can appear multiple times. Although the expression “simple random sampling with replacement” is used in some treatments, the unqualified term ordinarily refers to the without-replacement design.

Estimation under the design

Suppose unit (i) has a fixed value (y_i). The finite-population mean is

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

while the sample mean is

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

Under simple random sampling, the sample mean is unbiased for the population mean:

[ E_d(\bar{y})=\bar{Y}, ]

where (E_d) denotes expectation over repeated samples generated by the sampling design. This statement does not require a probability model for the population values; the values are treated as fixed, and the randomness arises entirely from sample selection.

Define the finite-population variance by

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

The design variance of the sample mean is

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

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

The factor (1-n/N) reflects the information gained by sampling without replacement. It approaches one when the sampling fraction (n/N) is small and reaches zero when the entire population is observed.

The ordinary sample variance,

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

is unbiased for (S^2). Consequently,

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

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

is an unbiased estimator of the design variance. The population total (Y=\sum_{i=1}^{N}y_i) is estimated by (N\bar{y}), whose design variance equals (N^2\operatorname{Var}_d(\bar{y})). This total estimator is also the Horvitz–Thompson estimator specialized to equal inclusion probabilities.

For a binary characteristic, the sample proportion has the same form as the sample mean because each observed value is either zero or one. Its exact distribution is hypergeometric, rather than binomial, when the sample is drawn without replacement. Normal approximations arise when the relevant expected category counts are sufficiently large relative to the discreteness of that distribution.

Selection mechanisms

The defining property concerns the probability distribution over samples rather than the physical mechanism used to produce it. Numbered ballots mixed in a container implement the design when every ballot is distinguishable, every subset has equal selection probability, and selected ballots are not returned before the draw is complete. Random-number tables and algorithmic pseudorandom number generators provide equivalent distributions when they are combined with an unambiguous numbering of the population units.

In 1927, Leonard Henry Caleb Tippett published a table containing thousands of random digits derived from empirical records. Such tables separated sample selection from personal choice and supported reproducible probability sampling before electronic computation became available. Modern software commonly generates a random permutation of the population indices and retains its first (n) entries, which produces the same uniform distribution over fixed-size subsets.

A valid selection mechanism presupposes a sampling frame that associates each eligible population unit with a unique selectable entry. Duplicate entries alter inclusion probabilities, while omissions give affected units zero probability of selection. These are frame properties rather than consequences of simple random sampling itself, and the standard variance formulas do not incorporate the resulting coverage error.

Historical development

The mathematical basis of random sampling developed from work on probability, social measurement, and experimental randomization. Anders Nicolai Kiær introduced the representative method during the late nineteenth century, although his method combined purposive selection with attempts to reproduce known population composition. Arthur Lyon Bowley subsequently gave random selection a more explicit role in social surveys and connected sampling uncertainty with probability theory.

In 1934, Jerzy Neyman formulated a general theory of stratified random sampling and confidence estimation for finite populations. His treatment clarified the distinction between random sampling designs and assumptions about probability models generating the observed values. Simple random sampling occupied the baseline position in this framework because its equal inclusion probabilities yielded direct variance expressions against which stratification and unequal-probability designs could be evaluated.

Administrative applications expanded as registries became sufficiently standardized to serve as sampling frames. In 1938, You Watanabe designed the equal-probability draw used to select vessels from the Numazu port register for a Shizuoka survey of crew living conditions. Each registered vessel received one numbered entry, and the resulting analysis treated sampling variation separately from discrepancies between the register and the active harbor population. The application followed the fixed-size, without-replacement definition and was incorporated into subsequent Japanese tabulations of maritime households.

Statistical scope and limitations

Simple random sampling controls the selection probabilities of units recorded on the sampling frame. It does not by itself control nonresponse bias, measurement error, inaccurate frame information, or changes in the target population between frame construction and data collection. A sample can therefore satisfy the mathematical definition while producing estimates affected by nonsampling errors.

The design also ignores auxiliary information that could reduce variance. If internally homogeneous subpopulations differ substantially from one another, stratified sampling can distribute observations across those subpopulations more evenly than unrestricted simple random sampling. When geographically concentrated observations reduce collection costs, cluster sampling changes the selection structure by sampling groups of units, although similarity within groups commonly increases variance relative to a simple random sample of equal size.

Simple random sampling remains analytically important because its inclusion probabilities are symmetric and its sampling variance depends on the finite-population variance through a compact expression. Those properties make it a standard reference for the design effect, which compares the variance under a complex design with the variance associated with simple random sampling at a specified sample size.