Lesson video

Hello, and thank you all for joining me, Mr. Gratin, in a sampling lesson where we will use the method of capture-recapture sampling to estimate the size of a population.

We'll look deeper into what makes a capture-recapture sample throughout the lesson, as we start by introducing what capture-recapture is.

Laura, who is getting far more confident with collecting a sample says, if she wants to find out how many rabbits everyone at school has as a pet, she could simply go up to everyone and ask them all.

How many rabbits do you each have? She has concluded that the 10 students at Oakfield Academy own a total of 10 rabbits.

But can she just go into a forest, and count all the rabbits there? Laura thinks that this is a little bit tricky.

How will she know if she's counted them all? So far, she's spotted three rabbits.

But rabbits are also known to move around a lot, so it will be hard to tell if she's counted the same rabbit twice or not.

There is definitely a fourth rabbit over there, but is this the same rabbit as before, or is it a different one? Do you count it and risk overestimating the population, or ignore it and risk underestimating the population instead? So she's spotted four or five rabbits so far, but some of them could still be hiding.

So she still doesn't know if she's counted the whole population.

I definitely know she has not spotted that one over there, did you? Is there ever a way to find the total number of rabbits that live in a forest? Well, it is tricky to get an exact accurate answer.

This is because the animals constantly move around, meaning you may miss some or count some twice.

Some animals might be in places hard for you to reach or spot, meaning that they will never be counted.

Some animals might move into or out of the habitat through migration, meaning the population may change.

And lastly, some animals may be born or die, meaning, again, the population may change.

Whilst the exact answer is rarely guaranteed, the population of an animal can be estimated through a process called capture-recapture.

The process of capture-recapture begins by capturing a number of an animal from their habitat, like so, and then marking them in some way.

For example, the animal could be tagged with a plastic band, or marked with paint or dye.

With these rabbits, we've placed some pink dye on their forehead.

Those animals are then released back into the habitat.

This is stage one of two, the capture stage of the capture-recapture process.

As you may have figured out, stage two is the recapture stage.

Over the next few days, we expect the animals in the habitat to mix back into the population, so that the marked animals mix in with the unmarked ones.

A few days later, the recapture is made, again, taking a random sample of the rabbits from the population.

The recapture does not need to have the same number of animals as the original capture.

In this recapture, we can check how many of the animals have the mark from the original capture.

So for this recapture of 14 rabbits, three of them have the pink dye on their foreheads.

Let's find out how we can use this information to estimate the population of rabbits in this habitat.

There are two ways of presenting the capture-recapture model, as a pair of equivalent fractions or as a ratio table.

Technically both representations are identical, and both will lead to the same estimate of the population.

Each representation just highlights a different part of the method.

Okay, let's start off with the equivalent fractions method.

On the first capture, we captured and marked seven rabbits out of a total population that we currently do not know the size of.

Because we don't know the size of this population, let's call the population size X.

On the recapture, we found that three of the rabbits that we recaptured were marked with the dye that we applied before, out of a total of 14 rabbits that we captured in this second part of the process.

This fraction represents the proportion of animals captured, compared to the whole population.

Seven rabbits captured out of the whole population, a population we currently do not know the size of.

Whereas this fraction represents the proportion of marked animals, compared to the total number recaptured.

Three marked rabbits out of a total of 14 rabbits captured.

Both fractions are equivalent to each other, as both fractions represent the same population of rabbits, with the right hand fraction representing a proportional sample of that population.

We can use proportional reasoning on the numerators.

The multiplier from the recapture to the capture is seven over three.

And so, X, the total population, is 14 times by the multiplier of seven over three, which is 32.

66, or 33 rabbits when round to the nearest whole rabbit.

Onto the ratio table representation, see if you can spot any similarities in our approach.

On the first capture, we captured and marked seven rabbits out of a total unknown population.

On the recapture, we found three marked rabbits out of a total of 14 captured rabbits.

We can also use proportional reasoning between the seven and the three on the top row of our ratio table.

But it is also possible to use proportional reasoning within the two values on our recapture column, giving us a multiplier of 14 over three.

Our population is therefore seven times by the multiplier of 14 over three, again, giving us 32.

66, or 33 whole rabbits after rounding.

Okay, for this check, 100 mealworms are captured and marked with a white dye.

During the recapture of 50 worms, three were marked with the white dye.

Pause here to think about or discuss, and then identify, which of these are correct for the original capture? For the original capture, there were 100 mealworms from a total unknown population that we can call X.

Same again, but this time, pause here to identify the correct fraction and ratio table column that represents the recapture.

A total of 50 worms were captured, of which three were marked with the dye.

By choosing either the pair of equivalent fractions or the ratio table, calculate a multiplier, and find an estimate for the number of mealworms in this colony.

Pause now to do this.

Regardless of which representation you chose, the correct calculation would have either involved 50 times 100, or 100 times by 50, whose product is then divided by three, which is 1,667 mealworms after rounding.

Whilst capture-recapture is most commonly seen whilst estimating the size of a population of animals, it can also be used to estimate any population or quantity, including humans or objects.

For example, the number of cars in a town centre is estimated by writing down the car registration number at the beginning and end of the working day, and seeing how many repeated registration numbers appear.

The capture involved noting down the registration number of 80 cars, whilst the recapture involved noting down 100 registration numbers, and identifying that 16 of them were repeats from the first capture.

We can represent this information in a ratio table, or pair of equivalent fractions.

The capture was noting down 80 registration numbers out of an unknown total population of cars.

The recapture was noting down 100 registration numbers, with 16 of them being repeats.

A multiplier across the recapture to the capture of 80 divided by 16 can be simplified to a multiplier of a multiplied by five.

Meaning that there were a total of approximately 100 times by five, or 500 cars in the town centre.

For this check, we have a concert.

The capture was giving 280 people flags to wave during the concert.

The recapture was during the concert, 50 people were asked to join in for a photo.

In this photo, seven people were holding flags.

Pause here to complete either the pair of equivalent fractions or the ratio table, and estimate the number of people who were at this concert.

And pause here to see if your calculations match the ones on screen.

And regardless of which method you chose, there were approximately 2,000 people at the concert.

Great stuff.

Onto the practise.

Pause here to try question one, which asks you to identify the pair of equivalent fractions and use them to estimate the population of wild pigeons in a local park.

And question two, pause here to identify the correct ratio table, and then use it to estimate the number of people in the town centre on Saturday.

And finally, question three, by choosing either the equivalent fraction or ratio table methods, conduct two different capture-recapture calculations to find out the change in population of ants in an ant colony over two years.

Pause now to do this.

Okay, here are the answers for question one.

128 pigeons were captured from an unknown population, X.

During the recapture, 24 out of 60 pigeons were wearing the bracelet.

An estimate for the number of pigeons is therefore, 60 times 128 divided by 24 equals 320 pigeons.

And for question two, 99 people were surveyed from an unknown number of people in the town centre.

During the recapture observation, 15 out of 130 people were wearing the wristband.

An estimate for the number of people in the town centre is therefore 130 times by 99 divided by 15 equals 858.

And finally, in 2023 there were 300 ants, whilst in 2024 there were 1,346 ants.

A 1,046 ant increase, that is a lot more ants.

Okay, now that we are familiar with what capture-recapture is, let's find out what makes an effective capture-recapture.

In order for a capture-recapture to produce a close estimate, one that's even remotely close to the real population, our capture methodology should collect a big enough sample during both the capture and recapture.

A larger sample size means a more accurate estimate.

This is because a lucky recapture with even one or two more marked animals than expected will have a massive impact on your estimation if the sample size is small.

But won't have anywhere near as much of an impact if the sample size is large.

A simple example could be that a representative capture is one marked out of 10 or, equivalently, 10 marked out of 100.

Two out of 10 is a massive proportional increase, whilst 11 out of 100 isn't much of an increase at all.

Furthermore, the animal should be marked with something that will last between the capture and recapture.

For example, if marking fish with paint or dye, it should be waterproof, otherwise it will wash off, decreasing the accuracy of your recapture.

It is also important to note the ethics behind your capture.

Be careful not to mark an animal with anything that will harm the animal, such as a tag that hurts the skin, or suffocates them, or something harmful to the environment, such as dye or paint that is toxic.

For this check, Laura brings rabbits in for capture by feeding them, then attaches a paper band around their neck.

Laura then recaptures 10 rabbits.

Oddly enough, she finds that none of them have the paper band around their necks.

Pause here to think about or discuss possible problems with Laura's capture-recapture methodology.

And pause here to identify which of these are likely to have had a bad influence on Laura's sample.

Paper is easily chewed through.

Also, some paper bands may have fallen off due to rainfall.

And, finally, Laura's sample size might have been too small.

If the population of rabbits is really, really large, then the first capture of eight just won't be big enough to be effective.

And finally, pause here to give advice to Laura on how she could have improved her capture-recapture methodology.

It's important to note that you probably should not use string, as rabbits can still easily chew through it, but waterproof paint can be helpful, as long as it is not toxic.

And definitely increase the number of rabbits captured and recaptured.

In addition to controlling your collection method, there are other assumptions that you need to make about the population, including assuming that the population does not change due to some animals being born or dying, or migrating in or out of the habitat.

In this diagram, we have some birds being born, some birds migrating away from the habitat, and some birds migrating into the habitat.

Also, you have to assume that both the capture and recapture are random samples of the population.

There is no point in doing a recapture if you're likely or guaranteed to find the exact same animals as in the original capture.

A way of doing this would be to take your capture and recapture from a range of locations in a habitat, rather than relying on one spot.

In this diagram, one squirrel from each group is captured.

Unlike this, don't take your whole sample from one tree or hole in the ground, reducing the possible captures down to one small group of animals from the whole population.

Assumptions for both the population staying the same, as well as a random sample being taken can be helped by choosing an appropriate timeframe between the capture and the recapture.

The timeframe must be long enough, so that you can assume the marked animals have mixed into the population.

So the recapture is random, but also that the timeframe isn't too long, that it gives enough time for the population to change due to births, deaths, or migration.

This is incredibly tricky to get right.

And the optimum timeframe, if one even exists, will change between different animals and different habitats.

Okay, for this check, what is your opinion about the timeframe between Jacob's capture and recapture of fish? Pause here to consider these options.

A month is too long, but why? Pause here to consider which of these explanations justify why a month is too long.

In a month, some fish will be born or die.

Some fish are captured by either humans or other animals for use as food.

And furthermore, the dye may wash off.

Even if the dye is waterproof, a month is a long time, especially for some fish species that shed their scales over time.

For this check, pause now to see if you can construct a ratio table or pair of equivalent fractions in order to estimate the size of the squirrel population.

Approximately 240 squirrels live in that habitat, but is 240 even a good estimate? Pause here to consider which of these assumptions must be true in order for the estimate to be accurate.

And the answer is all of them.

There are a lot of assumptions needed for the capture-recapture process to produce an effective estimate.

Okay, great work.

Let's bring together the practise of finding an estimate for a population from a capture-recapture, and the evaluation of whether the methodology is good or not in these practise questions.

Pause now to look through and evaluate the scenario in question one.

And pause here for the question two scenario.

Brilliant, here are the answers.

For question one, A, using one of the two capture-recapture techniques, we can say that there are approximately 840 pigeons in the park.

But the methodology was flawed, because the bracelets used to mark the pigeons could have been pecked off.

Meaning some of the pigeons in the recapture were also part of the original capture, but were not counted as they no longer had the bracelet.

For Part C, the assumption that the population did not change between the capture and recapture is probably not true, because some pigeons could have flown away in the two weeks between the capture and recapture.

And for question two, there are approximately 346 rabbits.

Laura's method was actually really good, because the recapture was a random sample of the population.

Four days is likely enough time for the captured rabbits to mix into the population, but not too long that the population of rabbits didn't change much in that time, as it isn't long enough for a lot of births, deaths, or migration to occur.

If dye did wash off some of the rabbits, then 346 is an overestimate of the population.

We can redo the capture-recapture calculations, with the recapture having more than 26 rabbits with red dye.

No matter what number greater than 26 that we choose, the result is an estimate lower than 346.

Pause now to modify your calculations for part A to try this for yourself.

Great work in learning and evaluating a new sampling technique in a lesson where we have considered that finding the exact population can be very difficult, but an estimate is possible through the use of capture-recapture sampling.

Assumptions about a big sample size, animal markings remaining between captures, and the population mixing, but remaining the same between two captures must all be true for a capture-recapture to give a good estimate.

Once again, thank you all so much for joining me in this sampling lesson.

I hope to see you all soon for some more maths, but until then, have an amazing rest of your day.