Mark and Recapture Calculator
Enter how many animals you marked, how many you caught the second time, and how many of those carried a mark, and get the Lincoln–Petersen and Chapman estimates of the population size with a standard error and two kinds of confidence interval. The working is shown, and the assumptions the method rests on are listed. Runs entirely in your browser. Free, no sign-up.
Results
480.2
Chapman estimate of N
500
Lincoln–Petersen estimate of N
91.9
Standard error of the Chapman estimate
164
Different animals actually seen (M + C − R)
20%
Marked share of the second sample (R ÷ C)
20.8%
Share of the population that is marked (M ÷ N)
95% confidence intervals for N
| Method | Lower | Upper |
|---|---|---|
| Normal approximation, Chapman estimate ± z × SE | 300 | 660.5 |
| Exact binomial (Clopper–Pearson on M ÷ N) | 328.5 | 841.3 |
Precision depends mostly on R: the relative standard error is about 1 ÷ √R, here about 25%.
M + C is below the estimate, so the Chapman estimator is not exactly unbiased. The bias is small when the expected number of recaptures, M × C ÷ N, is large.
The working
- N = M × C ÷ R = 100 × 80 ÷ 16 = 500
- N = (M + 1)(C + 1) ÷ (R + 1) − 1 = (100 + 1)(80 + 1) ÷ (16 + 1) − 1 = 480.24
- Var = (M + 1)(C + 1)(M − R)(C − R) ÷ ((R + 1)²(R + 2)) = 8,454.64, so SE = 91.95
Assumes a closed population (no births, deaths or movement between samples), no mark loss, equal chance of capture for every animal, and marks that do not change survival or behavior.
How to Use the Mark and Recapture Calculator
- 1. Enter M. This is the number of animals caught, marked and released in your first sample.
- 2. Enter C and R. C is the total number caught in the second sample, and R is how many of those already carried a mark. R cannot exceed M or C.
- 3. Choose the confidence level. 95% is the usual choice.
- 4. Read the estimate. The Chapman estimate is the headline figure; the Lincoln–Petersen figure is shown beside it for comparison. Check the notes under the intervals, which say when your data are too thin to rely on.
The Formulas
The idea is proportion. If M of the N animals in the population are marked, then the marked fraction M ÷ N should match the marked fraction R ÷ C in the second sample. Solving for N gives the Lincoln–Petersen estimator, which Petersen used on fish in 1896 and Lincoln applied to waterfowl banding in 1930. Chapman (1951) adjusted it to reduce its bias at small sample sizes:
Lincoln–Petersen
N = M × C ÷ R
Chapman
N = (M + 1)(C + 1) ÷ (R + 1) − 1
Variance of Chapman N
(M + 1)(C + 1)(M − R)(C − R) ÷ ((R + 1)²(R + 2))
Variance of Lincoln–Petersen N
M² × C × (C − R) ÷ R³
The Chapman variance is the estimator Seber gives in The Estimation of Animal Abundance and Related Parameters (2nd ed., 1982), and the standard error is its square root. The Chapman estimator is defined even when R = 0, where Lincoln–Petersen divides by zero. It is exactly unbiased when M + C is at least as large as the true N; otherwise it is only slightly biased as long as the expected number of recaptures is large. Report the result as a whole number of animals, and truncate rather than round if you follow the convention in the literature.
Worked Example
You mark 100 animals, release them, and later catch 80, of which 16 are marked. These are the calculator's default inputs.
- Lincoln–Petersen: 100 × 80 ÷ 16 = 500.
- Chapman: (101 × 81) ÷ 17 − 1 = 480.24. Its standard error is 91.95.
- Normal interval (95%): 480.24 ± 1.96 × 91.95 gives 300 to 660.5; the lower limit is also never allowed below the 164 different animals actually seen (100 + 80 − 16).
- Exact binomial interval (95%): the Clopper–Pearson limits for 16 of 80 convert to 328.5 to 841.3. It is wider on the upper side, which is where the uncertainty really lies.
With R = 16 the relative standard error is about 1 ÷ √16 = 25%. For the same M and C, doubling the recaptures to 32 would cut it to about 18%. To pin a 95% interval to roughly ±20% of N you need on the order of 100 recaptures, so it pays to catch more in the second sample rather than to mark more in the first.
| M | C | R | Lincoln–Petersen | Chapman | SE (Chapman) |
|---|---|---|---|---|---|
| 300 | 250 | 150 | 500 | 499.3 | 18.1 |
| 200 | 150 | 30 | 1,000 | 978.1 | 141.9 |
| 50 | 40 | 2 | 1,000 | 696 | 325.5 |
| 50 | 40 | 0 | undefined | 2,090 | 1,446 |
The Assumptions Behind the Method
- Closed population. No births, deaths, immigration or emigration between the two samples. Short intervals help; for a long study use an open-population model instead.
- Marks are not lost or missed. If marks fall off, R is too small and N is overestimated.
- Equal catchability. Every animal, marked or not, has the same chance of being caught. Trap-happy animals (which get caught again) make N too small; trap-shy animals make it too large.
- Marking does not change survival or behavior. A mark that makes animals easier for predators to see, or that you handle roughly, breaks the logic.
- Marked animals mix back into the population before the second sample, so the second catch is a random draw from it.
Frequently Asked Questions
- What is the Lincoln–Petersen formula?
- N = M × C ÷ R, where M animals are marked in the first sample, C are caught in the second, and R of those C are marked. It assumes the marked fraction of the second sample equals the marked fraction of the whole population.
- What is the difference between Lincoln–Petersen and Chapman?
- Chapman's version, N = (M + 1)(C + 1) ÷ (R + 1) − 1, is less biased with small samples and still gives an answer when no marked animals are recaptured. With many recaptures the two agree closely. Use Chapman unless you need to match a textbook answer.
- How many recaptures do I need?
- The relative standard error is roughly 1 ÷ √R, so 25 recaptures give about 20% and 100 recaptures give about 10%. A 95% interval is about twice that wide. With fewer than a handful of recaptures the estimate is very uncertain.
- Which confidence interval should I use?
- The normal interval is symmetric and fine when R is large. The exact binomial interval is asymmetric and holds up better when R is small, which is the common field situation. If the two disagree a lot, trust the exact one and collect more data.
- What if no marked animals are recaptured?
- Lincoln–Petersen is undefined, because you would divide by zero. The data say only that the population is large relative to your samples. Chapman gives a finite number, but with an enormous standard error, and the exact interval has no upper limit.
- Is this the same as capture-recapture used for other populations?
- The same two-sample logic is used outside ecology, for example to estimate undercount in registries. There the assumption of independent samples is often the weak point, because people found by one source are more likely to appear in another.