Home Statistical Dictionary Lincoln-Petersen Estimator

Lincoln-Petersen Estimator

The classical closed-form estimator of population size from a two-sample capture-recapture study, computed as the product of the two sample sizes divided by the number of individuals recaptured in both samples.

Sampling Theory & Survey MethodsIntermediateFree

In Plain English

The Lincoln-Petersen estimator is the actual formula behind basic capture-recapture population estimation, named for two independent developers, C.G.J. Petersen (fisheries, 1896) and Frederick Lincoln (waterfowl, 1930), who each arrived at the same simple, elegant idea. If you know how many animals you marked in the first capture, how many you caught in the second capture, and how many of the second capture were already marked, you can estimate the total population with just a bit of cross-multiplication, no calculus or iterative fitting required, and the intuition is straightforward: if a large fraction of your second catch turns out to be already-marked animals, the population probably isn't much bigger than what you've already sampled; if hardly any of the second catch is marked, there must be a lot more unmarked, uncaptured animals still out there.

Definition

The Lincoln-Petersen estimator computes an estimate of population size from a two-sample capture-recapture study as $\hat{N} = \frac{n _{1} n _{2}}{m}$ , where $n_{1}$ is the number marked in the first capture, $n_{2}$ is the total captured in the second sample, and $m$ is the number of individuals in the second sample found to be already marked; this follows directly from equating the sample recapture proportion $m / n_{2}$ to the population marked proportion $n_{1} / N$ and solving for $N$ . Because the raw estimator is biased upward in small samples (particularly when $m$ is small), especially when $m = 0$ (undefined) or very small, the Chapman correction, $\hat{N}_{C ha p man} = \frac{( n _{1} + 1 ) ( n _{2} + 1 )}{m + 1} - 1$ , is commonly used instead in practice, providing a nearly unbiased estimate even for small recapture counts, along with an associated variance formula for constructing confidence intervals around the estimated population size.

Formula

\hat{N} = \frac{n _{1} n _{2}}{m}

Notation

n_{1}

—Number of individuals marked in the first capture

n_{2}

—Total number captured in the second sample

m

—Number in the second sample already marked from the first capture

Properties

The raw Lincoln-Petersen estimator is undefined when m = 0 (no recaptures at all) and is upwardly biased in small samples more generally, which is why the Chapman correction, adding 1 to each of n1, n2, and m before dividing (and subtracting 1 from the result), is standard practice in applied capture-recapture studies rather than the uncorrected formula.
The Lincoln-Petersen estimator's validity depends entirely on the same core assumptions as the broader capture-recapture method, a closed population between the two captures and equal catchability for every individual regardless of prior capture history, violations of either assumption bias the resulting population estimate.
The Lincoln-Petersen estimator handles only the simplest two-sample case, modern capture-recapture studies with three or more capture occasions instead use more flexible log-linear or mixture models, which can test and relax the equal-catchability assumption in ways the simple two-sample Lincoln-Petersen formula cannot.

At a Glance

FormulaN-hat = n1 x n2 / m

Named forC.G.J. Petersen (1896, fisheries) and Frederick Lincoln (1930, waterfowl)

Small-sample bias fixthe Chapman correction, adding 1 to each count before dividing

Requiresthe closed-population and equal-catchability assumptions of capture-recapture

Last updated September 8, 2026← Back to the Dictionary