Petersen–Chapman population estimate

Wildlife biologists, fisheries scientists, and ecology students often need a closed-population abundance estimate from a two-sample mark–recapture study: mark M individuals, release them, capture C animals later, and count R marked recaptures. The classic Petersen ratio M times C over R is simple but biased for small samples and undefined when no recaptures occur. The Chapman modification replaces that ratio with N equals open parenthesis M plus one close parenthesis times open parenthesis C plus one close parenthesis over open parenthesis R plus one close parenthesis, then subtracts one, which reduces bias and remains defined when R is zero. This Petersen population adjusted calculator accepts the three integer counts, returns the Chapman point estimate, Seber variance and standard error, an approximate ninety-five percent interval floored at the number of unique individuals observed, and the classic Petersen value when R is positive—so homework checks, field notebooks, and automation pipelines share one deterministic contract.

What the Petersen–Chapman mark–recapture estimator measures

In a two-occasion closed-population mark–recapture design you capture a first sample, mark every individual in it, and release them so they remix with the unmarked animals. After mixing you capture a second sample of size C and count how many of those animals already carry a mark (R).

Under ideal assumptions—no births, deaths, immigration, or emigration between samples; marks not lost; equal capture probability; and random mixing—the proportion marked in the second sample estimates the proportion marked in the whole population, so population size N satisfies R over C approximately equals M over N, which rearranges to the classic Petersen estimator N equals M times C over R. That estimator is intuitive but has positive finite-sample bias and blows up when R is zero.

Chapman’s adjustment adds one to each count before dividing and then subtracts one from the ratio, yielding N hat equals open parenthesis M plus one close parenthesis times open parenthesis C plus one close parenthesis divided by open parenthesis R plus one close parenthesis, minus one. The adjustment is widely taught in wildlife and fisheries courses because it is nearly unbiased for moderate sample sizes under the same closed-population assumptions, and because it still returns a finite number when no marked animal is seen again.

This capability implements that Chapman point estimate exactly, together with the usual Seber sampling variance so you can report uncertainty alongside the point value. It does not fit open-population models, multi-occasion histories, or trap-dependence corrections; those need richer data and different estimators.

Where a Chapman population calculator earns its keep

Undergraduate and graduate ecology labs repeatedly ask students to turn mark and recapture tallies into an abundance estimate and a rough confidence interval. A deterministic calculator marks wrong the cases of fractional animals, negative counts, or R greater than M, instead of silently inventing a number.

Fisheries managers and wildlife technicians often need a quick closed-population check from a two-pass tagging event before investing in longer multi-occasion models: plug in M, C, and R, read the Chapman estimate and Seber standard error, and compare against the classic Petersen value when recaptures exist. Treat the result as a model-based estimate under closed-population assumptions, not as a census: if animals leave the study area, marks fall off, or capture probabilities differ between marked and unmarked individuals, the numerical output can be precise and still wrong for management decisions.