Chapman Estimator Calculators

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The Chapman estimator is a modified mark-recapture formula that provides an unbiased (or nearly unbiased) estimate of population size for small samples, improving on the original Lincoln-Petersen estimator. Lincoln-Petersen formula N̂ = (M × C)/R can be substantially biased (overestimates N) when the number of recaptures R is small. Chapman's modification: N̂ = (M+1)(C+1)/(R+1) − 1 reduces this positive bias. It is the recommended formula for mark-recapture studies, especially when sample sizes are small (R < 10). The Chapman estimator assumes the same conditions as Lincoln-Petersen: closed population, equal catchability, no mark loss.

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Chapman Formula

N̂ = (M+1)(C+1)/(R+1) − 1

M = animals marked and released in first sample; C = total animals captured in second sample; R = marked animals recaptured in second sample. This modification adds 1 to M, C, and R before the calculation, reducing the positive bias of the original Lincoln-Petersen formula N̂ = (M×C)/R when R is small.

Comparison with Lincoln-Petersen

When R is large (>10), Chapman ≈ Lincoln-Petersen. When R is small (< 5), Lincoln-Petersen can overestimate N by 10–50%; Chapman significantly reduces this bias. Example: M = 20, C = 30, R = 3. Lincoln-Petersen: N̂ = (20×30)/3 = 200. Chapman: N̂ = (21×31)/4 − 1 = 651/4 − 1 = 162.75 ≈ 163. The Chapman estimate is substantially lower and less biased.

Variance and Confidence Intervals

Approximate variance: Var(N̂) ≈ (M+1)(C+1)(M−R)(C−R) / [(R+1)²(R+2)]. 95% CI ≈ N̂ ± 1.96 × √Var(N̂). The CI is asymmetric and should be calculated using log-normal or Poisson methods for small samples.

Assumptions

Same as Lincoln-Petersen: closed population (no births, deaths, immigration, emigration); equal catchability of marked and unmarked animals; marks are retained and identified correctly; independence of captures between samples.

Glossary

Chapman Estimator
Modified mark-recapture formula: N̂ = (M+1)(C+1)/(R+1) − 1; reduces positive bias of Lincoln-Petersen when recaptures are few; the recommended formula for small-sample mark-recapture studies.
Lincoln-Petersen Estimator
N̂ = (M×C)/R; the classic mark-recapture formula; positively biased for small R; replaced by Chapman's modification in most applications.
Mark-Recapture
A population estimation method involving marking a sample of animals, releasing them, then recapturing a second sample and counting marked individuals; estimates N from the ratio of marked to total recaptured.

Frequently Asked Questions

The Chapman estimator N̂ = (M+1)(C+1)/(R+1) − 1 is a modified mark-recapture formula reducing the positive bias of Lincoln-Petersen N̂ = MC/R when recapture numbers are small. Lincoln-Petersen divides by R — when R is small (e.g., 1, 2, 3), this produces large, highly variable overestimates. Chapman's modification (adding 1 to M, C, and R) reduces this bias substantially. For R ≥ 10, both formulas give similar results. Chapman is the recommended formula for most mark-recapture studies.

Step 1: Capture and mark M individuals; release them. Step 2: After adequate mixing period, capture a second sample of C individuals; count R marked ones. Step 3: N̂ = (M+1)(C+1)/(R+1) − 1. Example: mark 50 fish (M=50); second sample catches 80 fish (C=80) including 10 marked ones (R=10): N̂ = (51×81)/11 − 1 = 4131/11 − 1 = 375.5 − 1 = 374.5 ≈ 375 fish. Compare Lincoln-Petersen: 50×80/10 = 400. For R=10, Chapman gives a slightly lower, less biased estimate.

Both Lincoln-Petersen and Chapman assume: (1) Closed population — no births, deaths, immigration, or emigration between the two samples. (2) Equal catchability — all individuals (marked and unmarked) have the same probability of being caught in the second sample. (3) No mark loss — marks must remain identifiable. (4) Random mixing — marked individuals must mix randomly with the population between samples. Violation of equal catchability (if marked animals become trap-shy or trap-happy) biases estimates. For populations that cannot be assumed closed (e.g., migratory species), open-population models (Jolly-Seber method) are used instead.

Approximate variance: Var(N̂) ≈ (M+1)(C+1)(M−R)(C−R)/[(R+1)²(R+2)]. SE = √Var(N̂). For large samples, approximate 95% CI = N̂ ± 1.96 × SE. For small samples, the distribution of N̂ is skewed right — use log-normal CI: lower = N̂/exp(1.96×CV); upper = N̂×exp(1.96×CV), where CV = SE/N̂. Or use software (Program MARK, R package 'Rcapture') for more accurate CIs. The precision of mark-recapture estimates depends heavily on the number of recaptures R — aim for R ≥ 20–30 for acceptably narrow confidence intervals.