Mark-Recapture Population Estimator
Solve the Lincoln-Petersen index N = (M×C)/R to estimate a wildlife population's total size from a mark-release-recapture study, plus the Chapman correction for small sample sizes. Two 3D diagrams compare a low-recapture-rate study (suggesting a large population) to a high-recapture-rate study (suggesting a small population), and charts show how the recapture count drives the population estimate.
Reviewed by the ToolNestr Editorial Team — July 2026
Low vs. high recapture rate
1. Low recapture rate
Few marked individuals turn up — implying a large total population.
2. High recapture rate
Many marked individuals turn up — implying a small total population.
Mark-recapture charts
How it works
The core idea in one line: if a known number of marked individuals are released back into a population, the fraction of marked animals that turn up again in a later sample reveals what fraction of the entire population those marked individuals actually represent — and that fraction is all you need to scale up to a total population estimate.
N = (M × C) / R
Lincoln-Petersen index — M=marked and released, C=second sample size, R=recaptured (marked) in second sample
N ≈ ((M+1)(C+1)/(R+1)) − 1
Chapman's correction — reduces bias when R is small
Mark-recapture works by assuming marked individuals mix back into the population randomly, so a second sample's proportion of marked-to-total individuals should mirror the whole population's proportion of marked-to-total individuals. Setting those two proportions equal (R/C = M/N) and solving for N gives the Lincoln-Petersen index, N = MC/R. Because this simple formula is known to overestimate population size when the recapture count is small, the Chapman correction adjusts each term slightly to remove that small-sample bias, giving a more reliable estimate whenever few marked individuals turn up in the second sample.
Worked example 1 — a standard mark-recapture study
Given: 50 fish are captured, marked, and released. A second sample later catches 60 fish, of which 12 are marked.
A recapture rate of 12/60 = 20% in the second sample implies the originally marked 50 fish represent about 20% of the total population.
Worked example 2 — a small sample needing the Chapman correction
Given: 10 individuals are marked and released. A second sample of 12 individuals includes 3 marked ones (a small recapture count).
The Chapman correction matters most when R (the recapture count) is small — with a recapture count of only 3, the simple formula noticeably overestimates the true population.
How recapture rate drives the population estimate
Fixed M=50 marked, C=60 second sample — only R (recaptures) changes.
| Recaptures (R) | Recapture rate | Population estimate |
|---|---|---|
| 6 | 10% | 500 |
| 12 ★ | 20% | 250 |
| 24 | 40% | 125 |
| 48 | 80% | 62.5 |
★ Reference row (worked example 1). A higher recapture rate always implies a smaller total population — the marked individuals are turning up more often because they make up a larger share of a smaller pool.
Where mark-recapture actually matters
🐟 Fisheries population assessment
Fisheries biologists use mark-recapture studies to estimate fish stock sizes, directly informing sustainable catch limits and conservation quotas.
🦋 Insect and butterfly population monitoring
Conservationists studying at-risk butterfly or insect populations use mark-recapture (often with small paint marks or tags) to track population trends over time.
🐻 Large mammal conservation
Wildlife biologists studying bears, big cats, or other elusive mammals often use camera-trap 'recapture' (identifying individuals from photos) as a non-invasive form of the same mark-recapture logic.
🦟 Disease vector and pest population studies
Public health researchers estimate mosquito or other disease-vector population sizes using mark-recapture to model disease transmission risk and evaluate control program effectiveness.
Common misconceptions
"Mark-recapture gives an exact population count, not an estimate."
It produces a statistical estimate based on proportions, not an exact count — the true population size could reasonably differ from the calculated N, especially with small sample sizes, which is exactly why confidence intervals are typically reported alongside the point estimate in real studies.
"The method works equally well no matter how small the recapture count is."
The basic Lincoln-Petersen formula becomes measurably biased when the recapture count (R) is small — this is precisely why the Chapman correction exists, and it's recommended whenever R falls below about 10.
"Mark-recapture assumes the population never changes between samples."
The classic Lincoln-Petersen method does assume a "closed" population (no significant births, deaths, immigration, or emigration between the two sampling events) — more advanced multi-sample mark-recapture models exist specifically to handle populations that change over time.
"Marking animals has no effect on the study's accuracy as long as you count carefully."
The method assumes marking doesn't change an animal's survival or likelihood of being recaptured — if marked individuals become more visible to predators, or become mark-shy or mark-prone during recapture, the estimate can be systematically biased regardless of how carefully the counting itself is done.
Formula sources & further reading
The formulas here are standard, traceable to:
- • OpenStax, Biology 2e — Chapter 45, "Population and Community Ecology" (free, peer-reviewed). openstax.org
- • Chapman (1951) — original small-sample bias correction paper, University of California Publications in Statistics.
- • Krebs, Ecological Methodology — Population estimation techniques chapter.
N = (M×C)/R (Lincoln-Petersen); N ≈ ((M+1)(C+1)/(R+1))−1 (Chapman correction). Assumes a closed population with equal capture probability. Results are rounded for display.
How to use this calculator
Enter marked count (M)
Individuals captured, marked, and released in the first sample.
Enter second sample and recaptures
Total second-sample size (C) and how many were already marked (R).
Compare both estimates
The simple Lincoln-Petersen result and the Chapman-corrected result display side by side.
Related tools
Frequently asked questions
What is mark-recapture?
Mark-recapture is a method for estimating a wildlife population's size: capture and mark a sample of individuals, release them, then take a second sample later and count how many are marked (recaptured) — the proportion marked in the second sample estimates the proportion marked in the whole population.
What is the Lincoln-Petersen index?
The Lincoln-Petersen index estimates total population size as N = (M×C)/R, where M is the number marked in the first sample, C is the total captured in the second sample, and R is the number of marked (recaptured) individuals found in that second sample.
Why does a low recapture rate suggest a large population?
If very few of the second sample turn out to be marked individuals, that implies the marked animals make up only a small fraction of a much larger overall population — the recapture rate directly reflects what proportion of the total population was originally marked.
What is the Chapman correction, and when is it needed?
The Chapman correction, N = ((M+1)(C+1)/(R+1)) − 1, adjusts for a known bias in the basic Lincoln-Petersen formula that becomes significant when the recapture count (R) is small — it's recommended whenever R is less than about 10.
What assumptions does mark-recapture rely on?
It assumes the population is closed (no births, deaths, immigration, or emigration between samples), marked individuals mix randomly back into the population, marking doesn't affect survival or recapture likelihood, and every individual has an equal chance of being captured.