Petersen capture-recapture (Higher tier)

Introduction to Petersen Capture-Recapture

Have you ever wondered how scientists know there are exactly 400 fish in a lake or 1,200 squirrels in a forest? They can't exactly ask the animals to line up and be counted! Instead, they use a clever statistical method called Petersen capture-recapture.

This technique is a vital part of the "Quality assurance and estimation" section of your course. It allows us to estimate the size of a whole population by looking at a small, marked sample. Because this is a Higher tier topic, we don't just learn the formula; we also look at the "rules" (assumptions) that make the estimate reliable.

How the Process Works

The method follows the statistical enquiry cycle in three clear steps:

1. The First Capture: You catch a group of individuals from the population. You count them (\(n_1\)), mark them in a way that doesn't hurt them, and release them back into the wild.
2. The Waiting Period: You wait long enough for the marked individuals to mix randomly with the rest of the population.
3. The Recapture: You catch a second group (\(n_2\)). You count how many in this second group are already marked (\(m\)).

The Petersen Formula

The logic is simple: the proportion of marked individuals in your second sample should be the same as the proportion of marked individuals in the entire population.

To find the total population estimate (\(N\)), use this formula:

\( \text{Estimated Population } (N) = \frac{\text{size of 1st sample } (n_1) \times \text{size of 2nd sample } (n_2)}{\text{number of marked individuals in 2nd sample } (m)} \)

Don't worry if this seems tricky! Just remember: Multiply the two sample sizes together, then divide by the "recaptured" (marked) number.

A Worked Example

A scientist wants to estimate the number of beetles in a field.

1. She catches and marks 60 beetles (\(n_1 = 60\)).
2. She releases them and returns a week later.
3. She catches a second sample of 80 beetles (\(n_2 = 80\)).
4. Out of these 80, 15 are marked (\(m = 15\)).

\( \text{Population Estimate} = \frac{60 \times 80}{15} \)
\( \text{Population Estimate} = \frac{4800}{15} \)
\( \text{Population Estimate} = 320 \)

Key Takeaway: The scientist estimates there are 320 beetles in the field.

The Vital Assumptions (Higher Tier Only)

For this estimate to be accurate, we have to assume several things are true. In your exam, you are often asked to comment on the reliability of an estimate based on these assumptions:

1. The population is "Closed": We assume no animals were born, died, moved into the area (immigration), or left the area (emigration) between the two captures.
2. Random Mixing: We assume the marked animals have mixed thoroughly and randomly with the rest of the population before the second sample is taken.
3. The Marks Stay Put: We assume the marks haven't fallen off, rubbed away, or been lost.
4. No Harm to the Animal: We assume the marking process doesn't make the animal more likely to die (e.g., making them easier for predators to see) or less likely to be caught again.
5. Equal Probability of Capture: Every individual in the population has the same chance of being caught in both samples.

Quick Review: What could go wrong?

Sample Size and Reliability

As with most things in Statistics, sample size matters. If your samples are very small, a tiny change in the number of recaptured animals (\(m\)) can cause a massive change in the total estimate. Larger samples generally lead to more reliable estimates and help mitigate the impact of random fluctuations.

Common Pitfalls to Avoid

1. Mixing up the numbers: Always double-check which number is the "marked" number. It is always the smallest number of the three (\(m\)).
2. Forgetting context: If an exam question mentions a hole in a fence or a disease in the population, they want you to explain why the "Closed Population" assumption is broken.
3. Rounding too early: Carry out your multiplication first, then divide. Only round your final answer to the nearest whole number (since you can't have half an animal!).

Summary Checklist

• Do you know the formula? \( N = \frac{n_1 \times n_2}{m} \)
• Can you list the 5 assumptions? (Closed population, random mixing, permanent marks, no harm, equal chance of capture).
• Can you explain why an estimate might be unreliable? (e.g., "The estimate may be too high because some animals might have lost their tags").

Note: For more on how sample sizes affect other types of estimations, see the chapter on "Sample size, reliability and replication".