Home Statistical Dictionary Sampling without Replacement

Sampling without Replacement

A sampling scheme in which each selected unit is removed from the population before the next draw, so no unit can be selected more than once and draws are not statistically independent.

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In Plain English

Sampling without replacement means once a unit is picked, it's set aside and can't be picked again, the standard way most real-world surveys and experiments actually work, since there's rarely a reason to interview or measure the same person twice. Because each draw shrinks the remaining pool, the draws aren't quite independent of each other the way they'd be with replacement, but this actually makes the resulting sample more efficient, not less, than an equivalent with-replacement sample, since the design is guaranteed never to waste effort on a unit already selected.

Definition

Sampling without replacement selects each unit from the population without returning it, so the population available for subsequent draws shrinks by one unit each time and no unit can appear more than once in the final sample of size $n$ drawn from a population of size $N$ ; the resulting draws are not marginally independent (the probability a given unit is selected on the second draw depends on which unit was removed on the first), though under simple random sampling every unit still has equal marginal inclusion probability $n / N$ . The variance of the without-replacement sample mean includes a finite population correction (FPC) factor, $Var (\overset{x}{ˉ}) = (1 - n / N) \frac{s ^{2}}{n}$ , which is strictly smaller than the corresponding with-replacement variance $s^{2} / n$ whenever $n > 1$ , reflecting the efficiency gain from never redrawing an already-sampled unit; as the sampling fraction $n / N$ approaches 0 the FPC approaches 1 and the two variances converge, but as $n / N$ approaches 1 (sampling close to a full census) the FPC approaches 0, correctly reflecting that a census leaves almost no sampling uncertainty at all.

Formula

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

Notation

1 - n / N

—Finite population correction (FPC) factor

n / N

—Sampling fraction

Properties

Sampling without replacement is more statistically efficient than sampling with replacement for the same sample size, because it never re-selects a unit already drawn, this is reflected directly in the smaller variance produced by the finite population correction factor (1 - n/N) < 1.
The finite population correction becomes practically important, and should not be ignored, when the sampling fraction n/N is not small, for example sampling 200 out of 1,000 units (n/N = 0.2); it becomes negligible and is routinely dropped when sampling from a very large population, for example a national survey sampling a few thousand out of hundreds of millions.
Despite individual draws technically not being independent under without-replacement sampling, the marginal inclusion probability of any single unit under simple random sampling is still exactly n/N, the same as under with-replacement sampling, it is only the joint probabilities of multiple units being selected together that differ between the two schemes.

At a Glance

Key propertyno unit can be selected more than once; the pool shrinks with each draw

Variance formulaincludes the finite population correction (1 - n/N), smaller than with-replacement

Standard practicethe default for essentially all real-world probability sample surveys

FPC near 1when n/N is small, effectively identical to with-replacement sampling

Last updated September 8, 2026← Back to the Dictionary