Capture-Recapture Calculators

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Capture-recapture (also called mark-recapture) is a statistical method for estimating the total size of a population by sampling it twice — marking individuals in the first sample, releasing them, then counting how many marked individuals appear in a second sample. The proportion of marked individuals recaptured in the second sample reflects what fraction of the whole population was initially marked, allowing the total population size to be estimated. The Lincoln-Petersen method is the simplest and most widely used approach, applied in wildlife ecology, fisheries, epidemiology, and even database record linkage.

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What Is Capture-Recapture?

Capture-recapture (mark-recapture) estimates population size (N) without counting every individual. The approach:

  1. Capture a sample of individuals, mark them (tags, paint, radio collars, bands), and release
  2. Allow sufficient time for marked individuals to mix randomly with the population
  3. Capture a second independent sample and count total captured (n) and number already marked (m)
  4. Estimate N from the ratio of marked to total in the second sample

Lincoln-Petersen Estimator

The standard formula (Peterson 1896, Lincoln 1930):

N = (M × C) / R

Where:

  • M — number marked and released in first sample
  • C — total captured in second sample
  • R — number of marked individuals recaptured in second sample

The Chapman modification (less biased with small samples):

N = ((M + 1)(C + 1) / (R + 1)) − 1

Worked Example

A wildlife team captures and tags 120 trout (M = 120). Two weeks later, they capture 95 trout (C = 95) and find 18 are tagged (R = 18):

N = (120 × 95) / 18 = 11,400 / 18 = 633 trout

95% confidence interval: N ± 1.96 × SE, where SE can be approximated.

Key Assumptions

  • Closed population: No births, deaths, immigration, or emigration between sampling occasions
  • Equal catchability: All individuals have equal probability of capture
  • Random mixing: Marked individuals mix randomly with unmarked before second capture
  • Marks are not lost: Tags are permanent and not overlooked
  • Marking does not affect survival or catchability

Open Population Models

Real populations have births, deaths, and movement. The Jolly-Seber model extends mark-recapture to open populations with multiple sampling occasions, estimating survival rates, birth rates, and population size simultaneously. The Cormack-Jolly-Seber (CJS) model estimates apparent survival from multi-occasion capture histories.

Applications Beyond Wildlife

  • Epidemiology: Estimating undiagnosed disease cases from multiple data sources
  • Database research: Estimating total records from overlapping databases
  • Human rights: Estimating conflict casualties from multiple witness accounts

Glossary

Lincoln-Petersen Estimator
The standard mark-recapture formula for estimating population size: N = (M × C)/R. Where M = initially marked, C = second sample size, R = recaptures. Assumes closed population, random mixing, and equal catchability.
Mark-Recapture
A wildlife population estimation method using two sampling events. Individuals are marked in the first sample and released; the recapture rate in the second sample estimates total population size. Also used in epidemiology and database research.
Closed Population Model
A population model assuming no births, deaths, immigration, or emigration during the study period. Required for Lincoln-Petersen estimator validity. Open population models (Jolly-Seber, CJS) relax this assumption for populations with demographic change.

Frequently Asked Questions

N = (M × C) / R, where M = number marked and released in the first sample, C = total captured in the second sample, R = number of marked individuals recaptured in the second sample. The formula assumes that the proportion of marked individuals in the second sample equals the proportion of the population that was marked — allowing population size to be inferred from the recapture ratio.

The key assumptions are: (1) Closed population — no births, deaths, immigration, or emigration between the two sampling occasions; (2) Equal catchability — all individuals have the same probability of being captured; (3) Random mixing — marked individuals redistribute randomly throughout the population before the second capture; (4) Marks are not lost and can be identified; (5) Marking does not affect survival or capture probability. Violations of these assumptions bias the population estimate.

In epidemiology, capture-recapture estimates the total number of cases of a disease or condition by cross-referencing multiple incomplete data sources (e.g., hospital records, death certificates, disease registries). Individuals appearing in both sources are 'recaptures.' The Lincoln-Petersen formula estimates the total case count including those missed by all sources. This is used to estimate undetected HIV, hepatitis, injury, and other condition burdens in populations.

The Chapman modification is N = ((M+1)(C+1)/(R+1)) − 1. It corrects the negative bias of the Lincoln-Petersen estimator when sample sizes are small or when R is small — in these cases the basic formula tends to underestimate N. The Chapman estimator is less biased and is preferred when expected recaptures are fewer than ~10. When sample sizes are large, both methods give nearly identical results.