Mark-Release-Recapture (Lincoln Index) Formula | Derivation & Calculator

Core idea

Overview

This statistical method assumes that the proportion of marked individuals in a second sample is representative of the proportion of marked individuals in the entire population. It is a critical tool for ecologists to measure population density without needing to observe every individual in a wild habitat. The method relies on the assumption that the population remains closed during the study period, meaning no births, deaths, or migration occur.

When to use: Use when studying mobile animal populations where direct counting is impossible or unethical.

Why it matters: Provides essential data for conservation efforts, pest control management, and biodiversity monitoring.

Symbols

Variables

N = Total population size, $n_{1}$ = Number marked in first sample, n2 = Total number caught in second sample, m2 = Number of marked individuals in second sample

N

Total population size

Variable

n_{1}

Number marked in first sample

Variable

n 2

Total number caught in second sample

Variable

m 2

Number of marked individuals in second sample

Variable

Walkthrough

Derivation

Derivation of Mark-Release-Recapture (Lincoln Index)

The Lincoln Index estimates population size by assuming that the proportion of marked individuals in a second sample is representative of the proportion of marked individuals in the entire population.

The population size is constant (no births, deaths, or migration).
Marking does not affect the survival or behavior of the individuals.
The marks remain visible and are not lost during the study period.
Marked individuals redistribute themselves randomly throughout the entire population.

1

Defining the Proportion

We assume the proportion of marked individuals in the second sample (m2/n2) is equal to the proportion of marked individuals in the total population (n1/N).

\frac{n _{1}}{N} = \frac{m _{2}}{n _{2}}

Note: n1 is the total number caught and marked initially; N is the total population.

2

Rearranging for Population Size

By cross-multiplying the previous proportion equation, we isolate the total population N multiplied by the number of marked individuals recaptured.

N \times m_{2} = n_{1} \times n_{2}

Note: Ensure you distinguish between n2 (total caught in second sample) and m2 (only those with marks).

3

Final Formula

Dividing both sides by m2 isolates N, providing the standard Lincoln Index formula used to estimate the total population.

N = \frac{n _{1} \times n _{2}}{m _{2}}

Note: If m2 is very small, the estimate becomes less reliable due to sampling error.

Result

N = \frac{n _{1} \times n _{2}}{m _{2}}

Source: AQA A-Level Biology Specification (3.7.1: Populations in ecosystems)

Free formulas

Rearrangements

Solve for $n_{1}$

Make $n_{1}$ the subject

n_{1} = \frac{N \times m _{2}}{n _{2}}

Isolate $n_{1}$ by multiplying both sides by $m_{2}$ and dividing by $n_{2}$ .

Difficulty: 2/5

Solve for $n_{2}$

Make $n_{2}$ the subject

n_{2} = \frac{N \times m _{2}}{n _{1}}

Isolate $n_{2}$ by multiplying both sides by $m_{2}$ and dividing by $n_{1}$ .

Difficulty: 2/5

Solve for $m_{2}$

Make $m_{2}$ the subject

m_{2} = \frac{n _{1} \times n _{2}}{N}

Rearrange the formula to solve for $m_{2}$ by multiplying by $m_{2}$ and dividing by N.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Why it behaves this way

Intuition

Imagine a giant bag of marbles. You pull out a handful and mark them (n1). You mix them back into the bag. Later, you pull out a second handful (n2). The fraction of marked marbles you find in your second handful (m2/n2) is assumed to be the same as the fraction of marked marbles in the entire bag (n1/N). It is a ratio-scaling problem: if 10% of your sample is marked, you assume 10% of the entire population is marked.

N

Total population size

The unknown total number of individuals living in the entire habitat.

n 1

Number of individuals caught and marked in the first sample

Your 'known' group; these are the individuals you have tagged for future identification.

n 2

Number of individuals caught in the second sample

Your 'snapshot' of the population, used to see how many previously marked individuals are present.

m 2

Number of marked individuals recaptured in the second sample

The overlap; these are the individuals that appear in both samples, serving as the link to estimate the total.

Signs and relationships

n1 ×n2: This represents the scaling factor of the two sampling events combined.
/: The division by m2 acts as a correction factor; it 'blows up' the sample size to represent the full population based on the proportion of recaptures.

One free problem

Practice Problem

Practice Problem 1

An ecologist captures 20 beetles, marks them, and releases them. Later, they capture 20 beetles and find 5 are marked. Calculate the total population.

Number marked in first sample20

Total number caught in second sample20

Number of marked individuals in second sample5

Solve for: $N$

Hint: Plug the values into (20 * 20) / 5.

Practice Problem 2

In a study of small rodents, 45 were trapped and marked. After a week, 60 were trapped, of which 12 were marked. Estimate N.

Number marked in first sample45

Total number caught in second sample60

Number of marked individuals in second sample12

Solve for: $N$

Hint: N = (n1 * n2) / m2

Practice Problem 3

A researcher marks 100 fish. In a follow-up, they catch 80 fish, 4 of which were marked. What is the estimated total population?

Number marked in first sample100

Total number caught in second sample80

Number of marked individuals in second sample4

Solve for: $N$

Hint: Check your arithmetic carefully.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Ecologists tagging 50 butterflies in a meadow, then returning a week later to capture 40 butterflies, finding that 10 of them have the tag, allowing the estimation of the total butterfly population.

Study smarter

Tips

Ensure marked individuals are given enough time to redistribute throughout the population before sampling again.
Check that marking methods do not make animals more conspicuous to predators.
Remember the assumption that the population size is constant between the two sampling events.

Avoid these traps

Common Mistakes

Assuming the population is 'open' (high migration or birth/death rates) which invalidates the result.
Forgetting that the 'marked' individuals must be allowed to mix back into the population fully before the second capture.
Ignoring the potential for the mark to rub off or be lost during the time between captures.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The Lincoln Index estimates population size by assuming that the proportion of marked individuals in a second sample is representative of the proportion of marked individuals in the entire population.

Use when studying mobile animal populations where direct counting is impossible or unethical.

Provides essential data for conservation efforts, pest control management, and biodiversity monitoring.

Assuming the population is 'open' (high migration or birth/death rates) which invalidates the result. Forgetting that the 'marked' individuals must be allowed to mix back into the population fully before the second capture. Ignoring the potential for the mark to rub off or be lost during the time between captures.

Ecologists tagging 50 butterflies in a meadow, then returning a week later to capture 40 butterflies, finding that 10 of them have the tag, allowing the estimation of the total butterfly population.

Ensure marked individuals are given enough time to redistribute throughout the population before sampling again. Check that marking methods do not make animals more conspicuous to predators. Remember the assumption that the population size is constant between the two sampling events.

References

Sources

Begon, M., Townsend, C. R., & Harper, J. L. (2006). Ecology: From Individuals to Ecosystems.
AQA A-Level Biology Specification 3.7.1
Lincoln, F. C. (1930). Calculating waterfowl abundance on the basis of banding returns.
A-Level Biology (OCR/AQA/Edexcel) Specification: Populations and Ecosystems.
AQA A-Level Biology Specification (3.7.1: Populations in ecosystems)

Mark-Release-Recapture (Lincoln Index)

Overview

Variables

Derivation

Defining the Proportion

Rearranging for Population Size

Final Formula

Rearrangements

Intuition

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Simpson's Diversity Index

Frequently Asked Questions

Sources