Home Statistical Dictionary Simple Random Sampling

Simple Random Sampling

A sampling design in which every possible sample of a given size from the population is equally likely to be chosen, so every individual unit has the same known probability of selection.

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In Plain English

Simple random sampling is the textbook baseline every other sampling design gets compared against: you pick units purely by chance, with no stratifying, clustering, or unequal weighting, so nobody's more or less likely to end up in the sample than anyone else. It's the design that makes the classic formulas for standard errors and confidence intervals from an introductory course exactly correct, which is precisely why real surveys, which almost always use something more complicated, still describe their departures from it.

Definition

Simple random sampling (SRS) selects a sample of size $n$ from a population of size $N$ such that every possible subset of size $n$ has equal probability of selection, $(n N)^{- 1}$ ; equivalently, every individual unit has the same inclusion probability $π_{i} = n / N$ . SRS can be conducted with replacement (each draw is independent, the same unit can appear more than once) or without replacement (units are removed from the pool once drawn, the more common case in practice); the sample mean $\overset{x}{ˉ}$ is an unbiased estimator of the population mean $μ$ under either version, but the sampling variance differs, without-replacement SRS is more efficient because it cannot redraw the same informative unit twice.

Formula

Var (\overset{x}{ˉ}) = (1 - \frac{n}{N}) \frac{s ^{2}}{n}

Notation

N

—Population size

n

—Sample size

π_{i} = n / N

—Inclusion probability of unit i under SRS

1 - n / N

—Finite population correction factor

Properties

SRS requires a complete, accurate sampling frame that lists every unit in the population; in practice this is often the hardest requirement to satisfy, which is exactly why designs like cluster sampling and multistage sampling exist as practical compromises.
The sample mean under SRS is design-unbiased for the population mean regardless of the population's underlying distribution, this design-based property, not a distributional assumption, is what justifies the standard formulas taught in introductory statistics.
SRS ignores any known structure in the population (geography, demographic strata, natural clusters); when such structure exists and is correlated with the variable of interest, stratified sampling or cluster sampling can be more efficient or more practical than SRS for the same total sample size.

At a Glance

Selection ruleevery unit has equal inclusion probability n/N

With vs. without replacementwithout-replacement is standard practice and more efficient

Rolethe theoretical baseline design every real-world sampling scheme is compared to

Requiresa complete sampling frame listing every population unit

Last updated September 8, 2026← Back to the Dictionary