Lincoln-Petersen Population Calculator
Mark-recapture is the standard field method for estimating the size of an animal population that cannot be counted directly. You capture and mark a first sample of animals, release them, then later take a second sample and count how many carry marks. The proportion of marked animals in the second sample reflects the proportion marked in the whole population, which lets you back out the total. This calculator returns both the classic Lincoln-Petersen index and the bias-corrected Chapman estimator, which is the recommended default for most studies.
Mark-recapture formula
Lincoln-Petersen: N = (M * C) / R
Chapman: N = ((M + 1)(C + 1) / (R + 1)) - 1
Recapture proportion = R / C
Chapman variance = ((M+1)(C+1)(M-R)(C-R)) / ((R+1)^2 (R+2))
Standard error = sqrt(variance)
M is the number marked in the first sample, C the total caught in the second sample, and R the number of those already marked. The Chapman estimator corrects the upward bias of the basic index and is the recommended default.
Using the estimate well
- The Chapman estimator is preferred for small samples and remains valid when no recaptures occur.
- Closed-population assumptions matter: keep the interval between samples short relative to birth, death, and migration rates.
- Marking must not change capture probability; trap-happy or trap-shy behaviour biases results.
- More recaptures shrink the standard error; aim for a meaningful number of marked animals in the second sample.
- The approximate standard error gives a sense of precision but is not a substitute for a formal confidence interval.
Mark-recapture: frequently asked questions
What is the Lincoln-Petersen estimator?
The Lincoln-Petersen index estimates total population size N from a two-sample capture-recapture study. You mark M animals in the first sample, capture C animals in the second sample, and observe R of them already marked (recaptures). The estimate is N = (M times C) / R. It assumes a closed population, no births, deaths or migration between samples, and that marks are not lost.
What is the Chapman estimator and why use it?
The Chapman estimator is a bias-corrected version of Lincoln-Petersen: N = ((M+1)(C+1) / (R+1)) - 1. The basic Lincoln-Petersen formula is biased upward, especially with small samples or few recaptures. The Chapman estimator is nearly unbiased when (M+C) is greater than or equal to N, and it remains defined when R = 0, so it is the recommended default for most field studies.
What are the key assumptions of mark-recapture?
The standard assumptions are: the population is closed (no births, deaths, immigration or emigration during the study); every animal has an equal chance of capture; marking does not affect recapture probability or survival; marks are permanent and correctly recorded; and marked animals mix randomly back into the population before the second sample.
What does the recapture count R need to be?
More recaptures give a more precise estimate. As a rule of thumb, studies aim for enough marked animals and a large enough second sample that R is at least 7 or more. With the basic Lincoln-Petersen formula, R must be greater than zero or the estimate is undefined; the Chapman estimator avoids this by adding 1 to the denominator.
Can this be used for fish, insects, or birds?
Yes. Mark-recapture is widely used by fisheries agencies, wildlife managers, and entomologists. The method applies to any countable, individually markable organism in a closed population over the study window. Marks include tags, bands, clips, dyes, or natural markings. The math is identical regardless of taxon.
Official sources
- U.S. Geological Survey: USGS science and wildlife population methods.
- U.S. National Library of Medicine: NCBI biostatistics literature on capture-recapture.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.