Estimation and Petersen Capture-Recapture

Counting the uncountable

How many fish are in a lake? You cannot catch and count every one. Instead, statisticians estimate: they study a sample and use it to work out a value for the whole population. One neat method is capture-recapture: catch some, mark them, let them go, then catch a second sample and see how many are marked. This module builds that method, its formula and the assumptions it depends on.

Sampling and estimate words

Estimation has a precise vocabulary. Get these exact.

Population to estimate

Bigger samples

What generally happens to a population estimate as the sample size increases?

• It generally becomes more reliable
• It always becomes less accurate
• It stays exactly the same
• It becomes impossible to calculate

The Petersen formula

For capture-recapture, the estimated population size is: N = (first sample x second sample) / number marked in the second sample In words: multiply the size of the two samples, then divide by how many marked individuals turned up in the second catch. On this Higher-tier topic the formula is not given - you must memorise it. Now try one.

Estimate the fish

To estimate the fish in a lake, 40 fish are caught, marked and released. Later, 50 fish are caught, of which 10 are marked. Using N = (first sample x second sample) / number marked, estimate the fish population.

How much can you trust it?

An estimate is only as good as the sample it came from. Sample size drives how much you can rely on the result.

Estimate the beetles

In a woodland, 60 beetles are caught, marked and released. A second sample of 80 beetles contains 12 marked ones. Estimate the beetle population.

What must be true?

Select the TWO assumptions the capture-recapture method relies on.

• The population is closed, with no births, deaths or migration between the samples
• The marks do not come off or affect survival
• The population must be very small
• Marked individuals deliberately avoid being recaptured

Why each assumption matters

Scale up a proportion

In a survey, 50 out of a sample of 200 people owned a pet. Estimate how many of a town's 8000 residents own a pet, by scaling up the sample proportion.

Is the estimate any good?

Three capture-recapture situations. Judge what has gone wrong or what to do.

• A biologist takes the second sample so soon that the marked fish have not mixed back into the lake. Why is the estimate unreliable?
• Between the two samples, a flood kills many animals and new young are born. Which assumption is broken?
• A student uses a tiny second sample of just three animals. What is the main problem?

Your turn: estimate

A conservationist catches, marks and releases 30 birds. In a second catch of 45 birds, 9 are marked. Estimate the bird population using the Petersen method, showing your working, and state ONE assumption the method relies on.

• Write the formula: N = (first sample x second sample) / number marked
• Substitute the numbers: 30 x 45 / 9
• Give the answer (150 birds)
• State one assumption, such as the population being closed between the two catches

Apply Petersen cleanly

Two things win the marks here. First, memorise and apply the Petersen formula cleanly - N = (first sample x second sample) divided by the number marked in the second catch - and always show your working so method marks are safe even if the arithmetic slips. Second, be ready to criticise an estimate: name the assumption that is broken (a population that is not closed, marks that wear off, poor mixing) or explain that a small sample makes the result unreliable. Estimation questions reward judgement, not just a number.