The Number on the Poster: Defending a Mark–Recapture Estimate
Students estimate the number of fish in a 0.4 ha desert pool. On day 1 they net 60 fish, clip a small fin, and release them. On day 2, 48 hours later, they catch 50 fish, of which 12 carry clips; they calculate 250 and plan to print that on a poster as the population size. Their field notes record that the largest fish were caught first and were the first to be marked, that two marked fish died in the holding bucket, that a heron was seen hunting at the pool overnight, and that the pool's surface area fell by about 20% between visits. The teacher asks them to state what the number does and does not represent.
1. Identify the assumptions of the method that are at risk here and the likely direction of bias in each case.
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The single mark–recapture estimate assumes a closed population, equal catchability of all individuals, no tag loss or tag-induced mortality, random mixing of marked animals, and a random second sample.
- Mortality (two marked fish died; heron predation) breaks the closed-population assumption. If losses fall on marked and unmarked alike, the estimate approximates the population on day 2; if marked fish survive worse (fin damage), the proportion of marked fish in the second sample is too low, so the estimate is biased upwards.
- Size-selective capture breaks equal catchability. If the biggest fish were marked and are also the most catchable on day 2, the second sample is too rich in marked fish, so the estimate is biased downwards. This is the most serious problem here.
- Tag loss would lower the recapture proportion and inflate the estimate.
- The shrinking pool does not bias the ratio directly, but it concentrates fish and changes catchability between visits, so the two samples are not drawn from the same population state.
2. Check their arithmetic, quantify the uncertainty, and say how the figure should be reported.
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Their arithmetic is correct: N = (M × C) / R = (60 × 50) / 12 = 250.
Uncertainty is large because only 12 recaptures were obtained. Approximate variance = M²C(C − R) / R³ = (3600 × 50 × 38) / 1728 ≈ 3960, giving a standard error of about 63 and a 95% interval of roughly 125–375. Chapman's modification, which is less biased for small recapture numbers, gives (61 × 51 / 13) − 1 ≈ 238.
Reporting: give about 240 with an interval of roughly 125–375, state that the interval rests on a normal approximation that is shaky with 12 recaptures, and describe the figure as an estimate for the day-2 population under stated assumptions - not as 'the number of fish in the pool'. Add the caveats about size-selective capture and mortality. Better still, present it as an index of abundance to be compared with later surveys.
3. Propose a redesigned protocol that reduces this bias and name an approach suited to an open population.
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Redesign: standardise capture effort across the size range (same net mesh sizes, same times, same sites, several short hauls rather than one), mark fish with a persistent individually coded tag (visible elastomer or PIT tag) rather than a fin clip that may affect swimming, hold fish briefly in aerated water and release promptly, and recapture after 24 hours rather than 48 so mortality and predation matter less. Include a same-day double-sample check on a subsample to test whether marked and unmarked fish are equally catchable; if they are not, correct the estimate or report a catch-per-unit-effort index instead.
For an open population, use a multiple-session method such as Schnabel or Jolly–Seber: mark and sample on several consecutive days so birth, death, immigration and emigration can be estimated separately rather than assumed absent. Sampling after the pool has stabilised, and taking fixed-effort samples, keeps the comparison meaningful.