Mark and Recapture Calculator: Lincoln-Petersen, Chapman and Schnabel Estimates
Estimate a wild population from mark-recapture data: the Lincoln-Petersen index and the less biased Chapman estimate with standard error and a 90, 95 or 99% confidence interval, or Schnabel and Schumacher-Eschmeyer estimates from several sampling occasions.
How it works
Lincoln-Petersen N = M·C ÷ R · Chapman N = (M + 1)(C + 1) ÷ (R + 1) − 1, Var = (M + 1)(C + 1)(M − R)(C − R) ÷ ((R + 1)²(R + 2)) · Interval = (M + C − R − ½) + (M − R + ½)(C − R + ½) ÷ (R + ½) × e^(±z·σ), σ² = 1/(R + ½) + 1/(M − R + ½) + 1/(C − R + ½) + (R + ½)/((M − R + ½)(C − R + ½)) · Schnabel N = Σ(C_t M_t) ÷ (Σ R_t + 1) · Schumacher-Eschmeyer N = Σ(C_t M_t²) ÷ Σ(R_t M_t)
Mark and recapture estimates the size of a population that cannot be counted directly. Animals are caught, marked and released; later a second sample is caught, and the share of that sample already carrying marks reflects the share of the whole population that was marked. Setting M/N = R/C and solving gives the Lincoln-Petersen index N = M·C ÷ R. With small samples that index overestimates, so Chapman’s version adds one to each count and is nearly unbiased once there are about seven or more recaptures; by convention it is truncated, not rounded. The calculator shows both, the Chapman standard error with a normal interval, and a log-scale confidence interval that keeps close to its stated coverage even for small samples. If you sampled more than twice, choose "Several samples" and enter each occasion’s catch and recaptures: the Schnabel method pools them as Σ(C·M) ÷ (ΣR + 1), where M is the number of marked animals at large before each sample, and the Schumacher-Eschmeyer regression estimate is shown as a check. All of these assume a closed population (no births, deaths or movement between samples), marks that are not lost, and equal catchability of marked and unmarked animals.
Worked example
If 40 fish are marked and released (M = 40) and a later catch of 50 (C = 50) contains 10 marked fish (R = 10), the Lincoln-Petersen estimate is 40 × 50 ÷ 10 = 200 fish. Chapman’s is 41 × 51 ÷ 11 − 1 = 189.09, truncated to 189, with a standard error of about 41.6. The 95% log-scale interval is 79.5 + (30.5 × 40.5 ÷ 10.5) × e^(±1.96 × 0.4015) = 79.5 + 117.64 × e^(±0.787), about 133 to 338 fish, and at least 80 different fish were handled.
Sources: Mark and recapture: Lincoln-Petersen, Chapman and confidence interval (Wikipedia)
Frequently asked questions
What assumptions does the method make?
The population does not change between samples (no births, deaths, immigration or emigration), marks are not lost or overlooked, marking does not change an animal’s chance of being caught, and marked animals mix randomly back into the population. Breaking any of these biases the estimate.
Should I use Lincoln-Petersen or Chapman?
Use Chapman for real data, especially when fewer than about 7 marked animals are recaptured: Lincoln-Petersen overestimates in small samples. With large samples the two agree closely. Many textbooks still teach Lincoln-Petersen, which is why it is the default headline here.
Why must recaptures be greater than zero?
The Lincoln-Petersen formula divides by R. If no marked animals turn up in the second sample, the population is probably much larger than (M + 1)(C + 1) − 1, or the marked animals did not mix back in; catch more animals or wait longer before resampling.
How many animals should I mark and recapture?
Enough to get at least 7 recaptures, and ideally 20 or more for a tight interval. As a rough guide, if you expect a population of N, marking and recapturing about 2√N animals each time gives around 4 recaptures; marking more narrows the interval faster than anything else.
What is the Schnabel method?
It extends Lincoln-Petersen to three or more sampling occasions. Each time you catch animals, count the marked ones, mark the new ones and release them all. The estimate pools every occasion: N = Σ(C × M) ÷ (ΣR + 1), where M is how many marked animals were at large before that sample.
Related calculators
Educational estimate. Check the method’s assumptions against your study design before relying on the figure.