Chapter 6 of 8

All 8 chapters
  1. 1Natural selection, readunderstood
  2. 2Evidence for evolution, readunderstood
  3. 3New species and extinction, readunderstood
  4. 4Classification, readunderstood
  5. 5The carbon and nitrogen cycles, readunderstood
  6. 6Measuring populations, readunderstood
  7. 7Interdependence, readunderstood
  8. 8Biodiversity and people, readunderstood

S4.3.6 Chapter 6

Measuring populations

How can you find how many daisies are in a field, or snails in a garden, without counting every one?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Estimate a population using quadrats and a transect
  • Use capture and recapture to estimate a number of snails or insects

To look after a place, or to study it, we need to know how many living things of each kind are there. Counting every one is usually too slow, and sometimes impossible. So scientists count a sample. This is a small part of the whole, and they scale it up.

For plants, and for animals that do not move, we use a quadrat. It is a square frame, often 1 m by 1 m, with an area of 1 m². You lay it on the ground and count the plants inside it. A plant that is outside the frame does not count.

A square frame, 1 m on each side, with the label 1 m above it and the words area 1 m² below it. Six plants stand inside the frame. Two more plants stand outside it, one beside the right edge and one below the bottom edge.1 marea 1 m²
A quadrat of 1 m by 1 m. Count only the plants inside the frame.

One quadrat is not enough. It may land on a thick patch or on a bare patch. So we place many quadrats, and we place them at random. Random means that every place has an equal chance. For example, you can pick the places with numbers written on slips of paper. Choosing a spot because it looks good would make the sample unfair.

Then we find the mean number of plants in a quadrat. We divide it by the area of one quadrat to get plants for each m². Last we multiply by the area of the whole field in m². This gives an estimate of the number of plants in the field.

A table with the columns Quadrat and Daisies. Quadrat 1 has 4. Quadrat 2 has 7. Quadrat 3 has 3. Quadrat 4 has 6. Quadrat 5 has 5.QuadratDaisies1427334655
Example data: daisies in five quadrats of 1 m², placed at random in a field.

Worked example

A field is 10 m by 10 m. Five quadrats of 1 m² hold 4, 7, 3, 6 and 5 daisies. Estimate the number of daisies in the field.

  1. The total in the quadrats is 4 + 7 + 3 + 6 + 5 = 25.
  2. The mean for each quadrat is 25 ÷ 5 = 5 daisies. A quadrat is 1 m², so this is 5 daisies for each m².
  3. The field has an area of 10 × 10 = 100 m², so 5 × 100 = 500 daisies.

Answer: About 500 daisies

A quadrat does not have to be 1 m². A smaller quadrat, 0.5 m by 0.5 m, has an area of 0.5 × 0.5 = 0.25 m². Then we must divide by 0.25, not by 1, to get the plants for each m².

A table with the columns Quadrat and Plants. Quadrat 1 has 2. Quadrat 2 has 4. Quadrat 3 has 3. Quadrat 4 has 1. Quadrat 5 has 5.QuadratPlants1224334155
Example data: plants in five quadrats, each 0.5 m by 0.5 m.

Worked example

A field has an area of 80 m². Five quadrats, each 0.5 m by 0.5 m, hold 2, 4, 3, 1 and 5 plants. Estimate the number of plants in the field.

  1. The total is 2 + 4 + 3 + 1 + 5 = 15, so the mean is 15 ÷ 5 = 3 plants for each quadrat.
  2. One quadrat has an area of 0.5 × 0.5 = 0.25 m².
  3. 3 plants in 0.25 m² is 3 ÷ 0.25 = 12 plants for each m².
  4. The field is 80 m², so 12 × 80 = 960 plants.

Answer: About 960 plants

Sometimes a living thing changes in number along a line. For example, grass may thin out from open ground into shade. To study this we use a transect. A transect is a line, such as a tape laid on the ground. Quadrats are placed at regular steps along it, and the plants in each are counted.

A tape marked in metres, with the title Distance along the line (m). The marks run from 0 to 40 in steps of 5. Five squares hang from the tape at those marks, labelled Q1, Q2, Q3, Q4 and Q5. They hold 12, 9, 5, 2, 0 plants.Distance along the line (m)051015202530354012Q19Q25Q32Q40Q5
A transect of 40 m with a quadrat every 10 m. The squares are symbols and are not drawn to scale. The numbers are the plants counted in each quadrat.

Quadrats suit plants, which stay where they are. Animals move, so they leave the square, or hide. For animals such as snails and insects we use capture and recapture. It works in four steps. First, catch a sample and mark each animal in a way that does not harm it. Then release them. Wait, so the marked animals mix with the others. Last, catch a second sample, and count how many have a mark.

If only a few of the second sample are marked, the population is large. If many are marked, the population is small. The estimate has three parts. Take the number marked in the first sample. Multiply it by the number in the second sample. Then divide by the number marked in the second sample.

Two panels joined by an arrow marked marked and released. A key names marked snail, drawn hatched, and snail not marked, drawn white. First sample shows 10 snails, all marked. Second sample shows 12 snails, of which 4 are marked. Under them is the sum 10 × 12 ÷ 4 = 30.marked snailsnail not markedFirst samplemarked and releasedSecond sample10 × 12 ÷ 4 = 30
Example numbers. The sum gives the estimate of the number of snails.

Worked example

Ravi catches 10 snails and marks all of them. He puts them back. Later he catches 12 snails and 4 have a mark. Estimate the number of snails.

  1. The marked animals in the first sample are 10. The second sample is 12, and 4 of it are marked.
  2. First multiply the marked in the first sample by the second sample: 10 × 12 = 120.
  3. Then divide by the marked animals in the second sample: 120 ÷ 4 = 30.
  4. The estimate is about 30 snails.

Answer: About 30 snails

The estimate is only fair if three things are true. The marks must not harm the animals or come off. The marked animals must mix back in with the others. And no animals are born, die or move in or out between the two samples.

Try it

Capture and recapture with paper slips

You need: a helper, a bag or a cloth sack, about 40 to 80 small slips of torn paper, a pencil.

  1. Your helper tears slips of paper and puts them in the bag, without telling you how many.
  2. Pick 10 slips from the bag. Put a pencil cross on each one. Put them back and shake the bag.
  3. Pick 10 slips again. Count how many have a cross.
  4. Work out the estimate: 10 × 10 ÷ the number with a cross. Then ask your helper for the real number.

What you will see: The estimate is close to the real number, but not the same. Try it three times and compare. Never use live animals for this. Use paper slips or beans only.

If you cannot try it, this is what happens.
A table with the columns Step and Slips. The rows are first sample marked, 10; second sample, 10; marked in second, 2; estimate, 50; real number, 46.StepSlipsfirst sample marked10second sample10marked in second2estimate50real number46

Try it

Count plants with a quadrat

You need: four straight sticks, each 1 m long, string to tie the corners, paper and a pencil, a lawn, a garden or a field, with the owner's leave.

Be safe: Mind the sharp ends of sticks and do not run with them. Wear a hat in the sun, and wash your hands after touching soil. Pick nothing and leave every plant and animal as you found it.

  1. Lay the four sticks in a square on the ground, and tie the corners with string. Each side is 1 m, so the area is 1 m².
  2. Choose 5 places at random. For example, write numbers on slips, draw them, and pace out that number of steps.
  3. In each place, count the plants of one kind inside the square, such as clover or daisies. Pick nothing.
  4. Work out the mean for each quadrat. Multiply it by the area of the whole patch in m².

What you will see: Each quadrat gives a different count, which is why many quadrats give a better estimate than one. Your answer is an estimate, not an exact number.

If you cannot try it, this is what happens.
A square frame, 1 m on each side, with the label 1 m above it and the words area 1 m² below it. Three plants stand inside the frame and one plant stands outside it, beside the right edge.1 marea 1 m²

A wrong idea many people have

Put the quadrat where the plants look thickest, so that you get a good count.

Why it feels right: A big count feels like a better, fuller count.

The test that breaks it: Place one quadrat on a thick patch and one on a thin patch of the same field. The counts differ a lot. The thick patch is not typical of the field.

What is true: A sample must be fair. Placing quadrats at random, and using many of them, gives a mean that stands for the whole field.

Key words

sample
a small part of the whole that is counted or measured, and then scaled up.
quadrat
a square frame of known size that is laid on the ground, to count the plants inside it.
random
chosen by chance, so that every place has an equal chance.
transect
a line across a habitat, along which quadrats are placed to show how living things change.
capture and recapture
a way to estimate the number of animals that move. Catch and mark some, release them, then catch again and count the marked ones.

Back to your guess

The answer is A: Count the daisies in several small areas chosen at random, find the average, and scale up to the whole field.

Counting everything is too slow, and often impossible. A fair sample, taken at random, gives a good estimate. Counting only where there are most would give too high a number.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Why should quadrats be placed at random?

    Tap your answer.

    1. Yes, that is right. If a person chooses the places, the choice can favour thick or thin patches. Random placement gives a fair sample.

    2. Not this one. Placing quadrats only where plants are thick would give a count that is too high. Random placement gives every place an equal chance. Try another.

    3. Not this one. Random quadrats still count only a sample. We scale the sample up to estimate the whole field. Try another.

    See the answer

    So that the sample is fair and does not favour places the person likes.

    If a person chooses the places, the choice can favour thick or thin patches. Random placement gives a fair sample.

    • If you chose B: Placing quadrats only where plants are thick would give a count that is too high. Random placement gives every place an equal chance.
    • If you chose C: Random quadrats still count only a sample. We scale the sample up to estimate the whole field.

  2. Look at the picture of a quadrat. How many plants are counted?

    A square frame, 1 m on each side, with the label 1 m above it and the words area 1 m² below it. Some plants stand inside the frame and some stand outside it, beside the edges.1 marea 1 m²
    Tap your answer.

    1. Not this one. That counts the plants outside the frame too. Only the plants inside the frame count. Try another.

    2. Not this one. That is the number of plants outside the frame. The plants inside the frame are counted. Try another.

    3. Yes, that is right. Seven plants stand inside the frame. The three plants outside it do not count.

    See the answer

    7

    Seven plants stand inside the frame. The three plants outside it do not count.

    • If you chose A: That counts the plants outside the frame too. Only the plants inside the frame count.
    • If you chose B: That is the number of plants outside the frame. The plants inside the frame are counted.

  3. Look at the transect. What does it show about the plant along the line?

    A tape marked in metres, with the title Distance along the line (m). The marks run from 0 to 40 in steps of 5. Five squares hang from the tape at those marks, labelled Q1, Q2, Q3, Q4 and Q5. They hold 12, 9, 5, 2, 0 plants.Distance along the line (m)051015202530354012Q19Q25Q32Q40Q5
    Tap your answer.

    1. Not this one. The counts are 12, 9, 5, 2 and 0, so they are not the same. Try another.

    2. Yes, that is right. The counts get smaller from Q1 to Q5, and the last quadrat holds none.

    3. Not this one. Read the counts from Q1 to Q5. They get smaller, not bigger. Try another.

    See the answer

    The number falls along the line.

    The counts get smaller from Q1 to Q5, and the last quadrat holds none.

    • If you chose A: The counts are 12, 9, 5, 2 and 0, so they are not the same.
    • If you chose C: Read the counts from Q1 to Q5. They get smaller, not bigger.

  4. Which is one of the things that must be true for capture and recapture to give a fair estimate?

    Tap your answer.

    1. Yes, that is right. If marks harmed animals or came off, the count would be wrong. So it would if the marked animals did not mix back in.

    2. Not this one. The two samples can be any size. What matters is that the animals are marked, mixed back in, and counted fairly. Try another.

    3. Not this one. The marked animals must mix back in with the others. If they stayed in one place, the second sample would not be fair. Try another.

    See the answer

    The marks do not harm the animals or come off, and the marked animals mix back in.

    If marks harmed animals or came off, the count would be wrong. So it would if the marked animals did not mix back in.

    • If you chose B: The two samples can be any size. What matters is that the animals are marked, mixed back in, and counted fairly.
    • If you chose C: The marked animals must mix back in with the others. If they stayed in one place, the second sample would not be fair.

  5. Some of the marks on the snails come off between the two samples. What happens to the estimate?

    Tap your answer.

    1. Not this one. Fewer marks are seen in the second sample. A smaller number marked makes the estimate larger, not smaller. Try another.

    2. Not this one. It is changed. The estimate divides by the number of marked animals found in the second sample, and that number is now too small. Try another.

    3. Yes, that is right. We divide by the marked animals in the second sample. If marks come off, that number is too small, so the estimate is too big.

    See the answer

    It is too high.

    We divide by the marked animals in the second sample. If marks come off, that number is too small, so the estimate is too big.

    • If you chose A: Fewer marks are seen in the second sample. A smaller number marked makes the estimate larger, not smaller.
    • If you chose B: It is changed. The estimate divides by the number of marked animals found in the second sample, and that number is now too small.

  6. Why do we use capture and recapture for snails, but quadrats for daisies?

    Tap your answer.

    1. Yes, that is right. A fixed frame suits things that stay still. Animals that move are counted by marking and recapturing.

    2. Not this one. Size is not the reason. Snails move, so they leave the frame or hide. Try another.

    3. Not this one. Quadrats are mostly used for plants and for animals that hardly move. They suit things that stay in place. Try another.

    See the answer

    Snails move about, so they cannot be counted in a fixed square. Daisies stay where they are.

    A fixed frame suits things that stay still. Animals that move are counted by marking and recapturing.

    • If you chose B: Size is not the reason. Snails move, so they leave the frame or hide.
    • If you chose C: Quadrats are mostly used for plants and for animals that hardly move. They suit things that stay in place.

  7. From an earlier chapter

    Rats carried on ships eat the eggs of birds that nest on the ground on an island, and the birds die out. Which cause of extinction is this?

    Tap your answer.

    1. Yes, that is right. The rats are living things that people carried to a new place, where they harm the local species.

    2. Not this one. Habitat loss means that the place the animals need is destroyed. Here the place is still there. The rats were carried in. Go to S4.3.3. Try another.

    3. Not this one. Eruptions are a natural cause. Here people carried the rats to the island, so it is an invasive species. Go to S4.3.3. Try another.

    See the answer

    An invasive species

    The rats are living things that people carried to a new place, where they harm the local species.

    • If you chose B: Habitat loss means that the place the animals need is destroyed. Here the place is still there. The rats were carried in. Go to S4.3.3.
    • If you chose C: Eruptions are a natural cause. Here people carried the rats to the island, so it is an invasive species. Go to S4.3.3.

  8. A field has an area of 50 m². Five quadrats of 1 m² hold 6, 4, 8, 5 and 7 plants. Estimate the number of plants in the field.

    See the answer

    About 300 plants

    The total is 6 + 4 + 8 + 5 + 7 = 30, so the mean is 30 ÷ 5 = 6 plants for each m². The field is 50 m², so 6 × 50 = 300.

  9. Look at the picture. Estimate the number of snails.

    Two panels joined by an arrow marked marked and released. A key names marked snail, drawn hatched, and snail not marked, drawn white. First sample shows 8 snails, all marked. Second sample shows 10 snails, of which 2 are marked.marked snailsnail not markedFirst samplemarked and releasedSecond sample
    See the answer

    About 40 snails

    The estimate is the marked in the first sample times the second sample, divided by the marked in the second sample. 8 × 10 ÷ 2 = 80 ÷ 2 = 40 snails.

Keep it

Copy this into your exercise book.

Count a random sample and scale up. Mean per quadrat, divided by its area, times the field area. A transect shows change along a line. Capture and recapture: marked first times second sample, divided by marked in the second.

Draw: A 1 m square frame with a few plants inside it and some outside, and the words area 1 m².

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (1.0 MB). It opens on any phone.