Understand the relationship
The reasoning behind the result
The overlap supplies the population information
M/N ≈ R/C
If the marked individuals have mixed through a closed population and the second sample is representative, the marked fraction of that sample approximates the marked fraction of the population. Rearranging gives the Lincoln–Petersen ratio estimate MC/R.
M is marked individuals available after the first occasion, C is all individuals in the second sample, and R is their marked subset. Recaptures cannot exceed either M or C. The distinct observed count is M + C − R and is reported separately from the estimated population.
Chapman's correction changes the small-sample arithmetic
Var(N̂) = (M+1)(C+1)(M−R)(C−R)/[(R+1)²(R+2)]
Adding one to the three counts and subtracting one from the resulting ratio gives the Chapman estimator. The displayed standard error is the square root of the conventional estimated variance under this model. It does not include bias from movement, mark loss or unequal capture probabilities.
A zero recapture count makes the Petersen estimate undefined. The Chapman expression remains finite, but zero observed overlap does not establish a defensible finite upper population bound. The output therefore identifies that case instead of treating the number as a reliable census.
The model can fail even when the arithmetic is correct
Births, deaths and migration between samples can violate population closure. Mark loss or missed marks changes observed overlap. Handling may make marked individuals more or less likely to be captured again, and capture probability may vary among individuals.
A standard error of zero in a complete-overlap boundary case is a property of the formula; it does not prove a complete census. This tool provides two-occasion arithmetic, not a multi-occasion, spatial or open-population capture model.
Follow the numbers
100 marked, 80 captured, 20 recaptured
- The marked fraction in occasion two is 20/80 = 0.25. Petersen gives 100/0.25 = 400 individuals.
- Chapman gives (101 × 81)/21 − 1 = 388.5714 individuals.
- The two occasions observed 100 + 80 − 20 = 160 distinct individuals; that observed count is not the estimated total.
The estimated total depends on the capture model. Fractional estimates are retained rather than implying that an exact integer population was measured.
Quick guide
How to use this calculator
- Enter the number actually marked and released after the first occasion.
- Enter total second-occasion captures and the subset recognised as marked.
- Check the overlap and distinct captured count before interpreting an estimate.
- Review the closed-population, mark-retention and catchability assumptions; sparse or zero overlap needs particular caution.
Calculation method
Calculation and interpretation
Estimate a closed population from marked individuals and their overlap with a second sample.
N̂Chapman = (M + 1)(C + 1)/(R + 1) − 1; N̂Petersen = MC/R
Worked example
100 marked, 80 captured, 20 recaptured
The estimated total depends on the capture model. Fractional estimates are retained rather than implying that an exact integer population was measured.
N̂Chapman = (M + 1)(C + 1)/(R + 1) − 1; N̂Petersen = MC/R
Supported inputs
Precision and limits
Sampling assumptions remain visible
These calculations summarize entered observations under the stated sampling model. They do not establish representative sampling, independent observations, perfect detection, habitat suitability or a population-management decision.
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