Home Statistical Dictionary Sampling with Replacement

Sampling with Replacement

A sampling scheme in which each selected unit is returned to the population before the next draw, so the same unit can be selected more than once and every draw is statistically independent.

Sampling Theory & Survey MethodsBeginnerFree

In Plain English

Sampling with replacement means you put each unit back after picking it, so it's eligible to be picked again, like drawing a card, noting it, and shuffling it back into the deck before the next draw. This makes every draw completely independent of every other draw, which is mathematically convenient (it's the assumption behind the bootstrap and behind most introductory probability formulas), even though most real surveys actually sample without replacement in practice, since interviewing the same person twice rarely adds anything useful.

Definition

Sampling with replacement selects each unit of the sample independently from the full population of size $N$ , returning every selected unit to the population before the next draw, so any given unit may appear in the sample more than once and each draw has an identical, independent probability distribution over all $N$ units. This independence is what makes sampling with replacement the natural setting for many classical probability results (each draw is an i.i.d. random variable), and it is the resampling scheme underlying the nonparametric bootstrap, which draws resamples of size $n$ with replacement from an observed dataset to approximate a statistic's sampling distribution; the variance of a with-replacement sample mean, $Var (\overset{x}{ˉ}) = σ^{2} / n$ , has no finite population correction term, unlike its without-replacement counterpart, because the population is effectively infinite from the perspective of any single draw.

Formula

Var (\overset{x}{ˉ}) = \frac{σ ^{2}}{n}

Notation

σ^{2}

—Population variance

n

—Sample size

Properties

Sampling with replacement makes draws i.i.d. (independent and identically distributed), which is the assumption underlying most classical probability theory and is exactly why bootstrap resampling, which needs draws to behave like fresh, independent samples from the data's empirical distribution, samples with replacement rather than without.
For a finite population, sampling without replacement is always at least as efficient as sampling with replacement for the same sample size, because it never wastes a draw re-selecting a unit already in the sample, the difference in efficiency shrinks toward zero as the population size N grows large relative to the sample size n.
When N is very large relative to n (a common situation in practice), sampling with and without replacement become nearly indistinguishable, since the probability of ever redrawing the same unit becomes negligible, this is why the finite population correction factor 1 - n/N is often dropped when n/N is small.

At a Glance

Key propertyeach draw is independent; the same unit can be selected more than once

Variance formulano finite population correction, unlike without-replacement sampling

Core applicationthe nonparametric bootstrap resamples with replacement from observed data

Practical surveysusually sample without replacement instead; each interview adds new information

Last updated September 8, 2026← Back to the Dictionary