Petersen capture-recapture formula Revision Notes for Edexcel GCSE Statistics

Petersen capture-recapture formula

What is the Petersen capture-recapture method?

The Petersen capture-recapture formula is a statistical technique used to estimate the size of large populations, particularly when it would be impossible or impractical to count every individual. This method is commonly used by ecologists and wildlife researchers to estimate animal populations in the wild.

The basic principle relies on the idea that if you mark a sample of individuals from a population, release them back, and then take another sample later, the proportion of marked individuals in your second sample should reflect the proportion of marked individuals in the entire population.

infoNote

The key insight behind this method is that marked individuals should be randomly distributed throughout the population after release, making the recapture sample representative of the overall population composition.

How the method works

The capture-recapture process follows these essential steps:

Step 1: Initial capture and marking

First, researchers catch a sample of individuals from the population (such as fish in a lake). Each individual is marked in a way that doesn't harm them or affect their behaviour, then released back into the population. It's crucial to record exactly how many individuals were marked and released.

Step 2: Recapture sample

After allowing time for the marked individuals to mix back into the general population, researchers take a second sample. In this recapture sample, they count both the total number of individuals caught and how many of these show the markings from the first capture.

Step 3: Using ratios to estimate population size

The method works because we can set up equivalent ratios. The ratio of marked individuals to the total in our recapture sample should equal the ratio of all marked individuals to the entire population.

For example, if we marked 4 fish initially, then recaptured 5 fish of which 2 were marked, we can write:

marked fish in sample : total fish in sample = marked fish in population : total population

This gives us: $2 : 5 = 4 : N$ (where $N$ is the total population we're trying to find)

The Petersen formula

The mathematical formula for estimating population size is:

$N = \frac{M \times n}{m}$

Where:

N = the estimated total population size (what we're trying to find)
M = number of individuals marked in the first capture and released
n = total size of the recapture sample
m = number of marked individuals found in the recapture sample

This formula comes from rearranging the ratio: $\frac{M}{N} = \frac{m}{n}$

Essential assumptions

For the Petersen capture-recapture method to give accurate results, several key assumptions must be met:

chatImportant

Critical Assumption - Closed Population

The population must be closed, meaning no individuals enter or leave the study area between the marking and recapture phases. This means no migration, births, or deaths should occur during the study period.

Equal catchability

All individuals in the population must be equally likely to be captured in both the initial and recapture samples. The marking process must not make individuals more or less likely to be caught again.

Marks remain visible

The marking system must be permanent and clearly visible during the recapture phase. Marks shouldn't fade, fall off, or become illegible.

infoNote

Representative Sampling Requirements

Both the initial and recapture samples must be large enough and random enough to be representative of the entire population. Small sample sizes can lead to significant errors in population estimates.

No behavioural changes

The marking and initial capture process must not change the behaviour of the marked individuals in ways that would affect their likelihood of being recaptured.

Worked example: Mice on an island

lightbulbExample

Worked Example: Estimating Mouse Population

Problem: A scientist wants to estimate the number of mice on an island. She captures 60 mice, marks them and releases them. Later she captures 40 mice, of which 6 were already marked. Find an estimate for the number of mice on the island.

Solution: Using the formula $N = \frac{M \times n}{m}$

Given information:

$M = 60$ (mice marked in first capture)
$n = 40$ (total mice in recapture sample)
$m = 6$ (marked mice found in recapture sample)

Substituting into the formula: $N = \frac{60 \times 40}{6} = \frac{2400}{6} = 400$

Answer: There are approximately 400 mice on the island.

Key assumptions for this study:

The marked mice do not die between the marking and recapture phases
The probability of being caught is the same for marked and unmarked mice (marking doesn't affect their behaviour)

Common exam tips and traps

infoNote

Remember to:

Always check that your answer makes sense - the population estimate should be larger than the number initially marked
State the key assumptions when asked
Show your working clearly, substituting values into the formula step by step
Round your final answer appropriately (usually to the nearest whole number for population estimates)

chatImportant

Common Mistakes to Avoid:

Mixing up the variables in the formula (make sure $M$ is the initial number marked, not the recapture sample size)
Forgetting that this gives an estimate, not an exact count
Not considering whether the assumptions are realistic for the given scenario

Summary

bookmarkSummary

Key Points to Remember:

The Petersen capture-recapture formula estimates population size using the principle that marked individuals should be recaptured in the same proportion as they exist in the total population
The formula is $N = \frac{M \times n}{m}$ , where $N$ is population size, $M$ is initially marked, $n$ is recapture sample size, and $m$ is marked individuals in recapture sample
The method relies on several crucial assumptions including a closed population, equal catchability, and permanent marking
Always check your answer makes sense - the estimated population should be larger than the initial number marked
This method gives an estimate, not an exact count, and is most useful for large populations that cannot be counted directly

Petersen capture-recapture formula (Edexcel GCSE Statistics): Revision Notes