Chapter 7 of 8

All 8 chapters
  1. 1Sorting living things, readunderstood
  2. 2Using and making keys, readunderstood
  3. 3Food webs, readunderstood
  4. 4Energy through a food chain, readunderstood
  5. 5Decomposers and recycling, readunderstood
  6. 6Competition and adaptation, readunderstood
  7. 7Counting living things, readunderstood
  8. 8People and ecosystems, readunderstood

S3.2.7 Chapter 7

Counting living things

How can anyone count the daisies in a whole field?

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 the number of plants in a field using a square frame
  • Explain why samples should be placed at random

Counting every plant in a big field would take days. So scientists count a small part and use it to estimate the whole. The small part is called a sample.

For plants, the sample is taken with a quadrat. A quadrat is a square frame made of sticks and string. Here the frame is 1 m on each side, so it covers 1 m².

Draw it first

Count plants with a frame

  1. A field drawn as a grid, built in three steps. First the grid, then the frames, then the counts. The field is drawn as 6 columns and 4 rows of squares, 6 m across and 4 m down. Each square is the size of one frame. 5 squares hold a frame, with the plants counted in each: 9, 4, 5, 4, 3. Labels: 6 m, 4 m, 9, 4, 5, 4, 3.6 m4 m

    1 Measure the field and mark it in squares the size of the frame. This field is 6 m by 4 m, so it has 24 squares.

  2. A field drawn as a grid, built in three steps. First the grid, then the frames, then the counts. The field is drawn as 6 columns and 4 rows of squares, 6 m across and 4 m down. Each square is the size of one frame. 5 squares hold a frame, with the plants counted in each: 9, 4, 5, 4, 3. Labels: 6 m, 4 m, 9, 4, 5, 4, 3.6 m4 m

    2 Put the frame down on 5 squares. Choose the squares at random.

  3. A field drawn as a grid, built in three steps. First the grid, then the frames, then the counts. The field is drawn as 6 columns and 4 rows of squares, 6 m across and 4 m down. Each square is the size of one frame. 5 squares hold a frame, with the plants counted in each: 9, 4, 5, 4, 3. Labels: 6 m, 4 m, 9, 4, 5, 4, 3.6 m4 m94543

    3 Count the plants inside each frame, and write the count in the frame.

Look closely at one frame. Count only the plants inside it. A plant lying across the string counts if its middle is inside. Use the same rule every time.

One square frame, 1 m on each side, on the ground with 7 small plants inside it and some more outside. Count only those inside: 7. Labels: 1 m.1 m
7 plants are inside this frame. The plants outside are not counted.

The next picture shows five frames placed at random. Each number is the plants counted in that frame.

See it three ways

With things

Write the numbers 1 to 24 on scraps of paper, one for each square. Fold them and put them in a cup. Shake the cup and draw 5 scraps without looking. Put a frame on each square you draw.

As a picture

Five frames placed at random. The field is drawn as 6 columns and 4 rows of squares, 6 m across and 4 m down. Each square is the size of one frame. 5 squares hold a frame, with the plants counted in each: 9, 4, 5, 4, 3. Labels: 6 m, 4 m, 9, 4, 5, 4, 3.6 m4 m94543

In words

Each square had the same chance of being drawn. Some frames fell on thick patches and some on thin ones. That is what a fair sample looks like. Throwing a pebble is not fair. Squares near you are hit more often, and far squares are never hit.

In symbols

9 + 4 + 5 + 4 + 3 = 25 plants in 5 frames

Now work out the mean. To find the mean, add the counts and divide by how many counts there are. The mean is the number of plants in a typical frame.

A table of the plants counted in each frame. Frame 1 has 9. Frame 2 has 4. Frame 3 has 5. Frame 4 has 4. Frame 5 has 3. The total is 25 and the mean is 5. Labels: frame, plants, 1, 2, 3, 4, 5, 9, 4, 5, 4, 3, total, mean, 25, 5.frameplants1924354453total25mean5
The counts from the five frames, with their total and mean.

Each frame covers one square. The field has 24 squares. If a typical square holds 5 plants, the field holds 5 times 24.

Worked example

Five frames hold 9, 4, 5, 4, 3 plants. The field is 6 m by 4 m, so it has 24 squares the size of the frame. Estimate the plants in the field.

  1. Add the counts: 9 + 4 + 5 + 4 + 3 = 25.
  2. Divide by the number of frames: 25 ÷ 5 = 5 plants in a typical frame.
  3. Multiply by the squares in the field: 5 × 24 = 120.

Answer: 120 plants

This is an estimate, not an exact count. Say that every plant had been counted, and the total was 125. The estimate of 120 is close. A different set of squares would give a slightly different estimate.

More frames make a better estimate. The graph shows the estimate after one frame, two frames and so on. It comes closer to the real total as more frames are counted.

A line graph of the estimate for the whole field after 1, 2, 3, 4 and 5 frames. The estimate is 216, 156, 144, 132, 120 plants. It starts high and comes down to 120. Labels: estimate, frames, 1, 2, 3, 4, 5, 0, 40, 80, 120, 160, 200, 240.0408012016020024012345framesestimate
The estimate after each frame. The real total is 125.

Now, why at random? Random means chosen by chance. Suppose you put every frame near the thick left edge. You would count a lot of plants and think the whole field is thick.

Five frames all put near the thick left edge: the four squares of the first column and one beside them. The field is drawn as 6 columns and 4 rows of squares, 6 m across and 4 m down. Each square is the size of one frame. 5 squares hold a frame, with the plants counted in each: 10, 9, 11, 9, 6. Labels: 6 m, 4 m, 10, 9, 11, 9, 6.6 m4 m1091196
All the frames sit near the thick left edge. The mean is 9, so the estimate is 216. That is far too high.

Try it

Count a patch

You need: four sticks of equal length, about 1 m each, string to tie them into a square, 24 scraps of paper and a cup, a pencil and paper, a grassy patch of known size, such as 6 m by 4 m.

Be safe: Look for glass, thorns and nests before you start, and keep away from any you find. Look, count and leave the plants as they are.

  1. Tie the sticks into a square frame. Pick one kind of plant to count, such as clover or daisies.
  2. Lay a 1 m stick along the long side and then the short side. Multiply the two counts to get the squares. Say it is 24. Number them 1 to 24 row by row from one corner. Write the numbers on scraps and put them in the cup.
  3. Shake the cup and draw 5 scraps without looking. Put the frame on each square you draw, and do not choose the square yourself.
  4. Count the plants of your kind inside the frame. Write the number. Do this 5 times.
  5. Find the mean. Multiply it by the number of squares in the patch.

What you will see: Each frame gives a different count. The mean smooths them out. Another time, with other draws, the answer will change a little but stay close.

If you cannot try it, this is what happens.
A bar chart of the plants counted in each of five frames. Frame 1 9, Frame 2 4, Frame 3 5, Frame 4 4, Frame 5 3. Labels: plants in frames 1 to 5, 1, 2, 3, 4, 5, 0, 2, 4, 6, 8, 10.024681012345plants in frames 1 to 5

A wrong idea many people have

Put the frames where the plants are thickest, so you do not miss them.

Why it feels right: You want a good count, and thick patches have plenty to count.

The test that breaks it: Look at the picture of frames near the left edge. The estimate is 216, but the real total is 125.

What is true: Choose the squares at random. Then thick and thin patches both get a fair chance. The mean is a fair guide to the whole field.

Key words

sample
a small part that you count or measure to learn about the whole.
quadrat
a square frame, here 1 m on each side, laid on the ground to count plants.
estimate
a careful guess worked out from a sample.
mean
the total of the counts divided by how many counts there are.
random
chosen by chance, so that every place has the same chance.

Back to your guess

The answer is C: They count the plants inside a few small frames, then use the counts to estimate the whole field.

Scientists count the plants in a few square frames and work out the mean. Then they multiply by the number of frames that fit in the field. It is an estimate, and a fair one if the frames are placed at random.

Check yourself

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

  1. Why do scientists count plants in a few frames, not in the whole field?

    Tap your answer.

    1. Yes, that is right. A sample is a small part of the whole. Counting it is quick, and it tells you a good deal about the whole field.

    2. Not this one. It can be counted, but it would take days. A few frames give a good estimate in an hour. Go to S3.2.7. Try another.

    3. Not this one. A person does the counting. The frame only marks off a small, known area to count. Go to S3.2.7. Try another.

    See the answer

    Counting all the plants would take too long, and a sample gives a good estimate.

    A sample is a small part of the whole. Counting it is quick, and it tells you a good deal about the whole field.

    • If you chose B: It can be counted, but it would take days. A few frames give a good estimate in an hour. Go to S3.2.7.
    • If you chose C: A person does the counting. The frame only marks off a small, known area to count. Go to S3.2.7.

  2. A student counts 3, 5, 6 and 2 plants in 4 frames. What is the mean number of plants in a frame?

    Tap your answer.

    1. Not this one. That is the total of the counts. To find the mean, divide the total by the number of frames. Go to S3.2.7. Try another.

    2. Yes, that is right. Add: 3 + 5 + 6 + 2 = 16. Divide by 4 frames: 16 ÷ 4 = 4.

    3. Not this one. You may have picked one of the counts. The mean is the total divided by the number of counts. Go to S3.2.7. Try another.

    See the answer

    4 plants

    Add: 3 + 5 + 6 + 2 = 16. Divide by 4 frames: 16 ÷ 4 = 4.

    • If you chose A: That is the total of the counts. To find the mean, divide the total by the number of frames. Go to S3.2.7.
    • If you chose C: You may have picked one of the counts. The mean is the total divided by the number of counts. Go to S3.2.7.

  3. The mean is 6 plants in each frame. The field has 30 squares the size of the frame. Estimate the plants in the field.

    Tap your answer.

    1. Yes, that is right. Each square holds about 6 plants and there are 30 squares. 6 × 30 = 180.

    2. Not this one. You have added the two numbers. Each of the 30 squares holds about 6 plants, so multiply. Go to S3.2.7. Try another.

    3. Not this one. Dividing 30 by 6 gives a number that means nothing here. Each square holds about 6 plants, so multiply. Go to S3.2.7. Try another.

    See the answer

    180 plants

    Each square holds about 6 plants and there are 30 squares. 6 × 30 = 180.

    • If you chose B: You have added the two numbers. Each of the 30 squares holds about 6 plants, so multiply. Go to S3.2.7.
    • If you chose C: Dividing 30 by 6 gives a number that means nothing here. Each square holds about 6 plants, so multiply. Go to S3.2.7.

  4. Why should the frames be placed at random?

    Tap your answer.

    1. Not this one. Finding them again does not matter. What matters is that no part of the field is favoured over another. Go to S3.2.7. Try another.

    2. Yes, that is right. Random choice gives thick and thin patches the same chance. The sample then matches the whole field.

    3. Not this one. Frames on thick patches would find more plants, but the estimate would be too high. A fair sample finds the plants a typical square holds. Go to S3.2.7. Try another.

    See the answer

    So that every part of the field has a fair chance of being counted

    Random choice gives thick and thin patches the same chance. The sample then matches the whole field.

    • If you chose A: Finding them again does not matter. What matters is that no part of the field is favoured over another. Go to S3.2.7.
    • If you chose C: Frames on thick patches would find more plants, but the estimate would be too high. A fair sample finds the plants a typical square holds. Go to S3.2.7.

  5. A student puts all five frames near the thick left edge. What goes wrong?

    Five frames all put near the thick left edge. The field is drawn as 6 columns and 4 rows of squares, 6 m across and 4 m down. Each square is the size of one frame. 5 squares hold a frame, with the plants counted in each: 10, 9, 11, 9, 6. Labels: 6 m, 4 m, 10, 9, 11, 9, 6.6 m4 m1091196
    Tap your answer.

    1. Yes, that is right. The frames sit where the plants are thickest. The mean is too high, so the estimate for the whole field is too high.

    2. Not this one. The left edge has more plants than the rest of the field. The counts are high, so the estimate is too high. Go to S3.2.7. Try another.

    3. Not this one. The frames are the same size, but they all sit on the thickest part. They do not stand for the whole field. Go to S3.2.7. Try another.

    See the answer

    The estimate is too high.

    The frames sit where the plants are thickest. The mean is too high, so the estimate for the whole field is too high.

    • If you chose B: The left edge has more plants than the rest of the field. The counts are high, so the estimate is too high. Go to S3.2.7.
    • If you chose C: The frames are the same size, but they all sit on the thickest part. They do not stand for the whole field. Go to S3.2.7.

  6. From an earlier chapter

    Plants in a crowd compete. What do they compete for?

    Tap your answer.

    1. Yes, that is right. A plant needs light, water, space and nutrients. In a crowd there is less of each for every plant.

    2. Not this one. Plants make their own food from light. They compete for light, water and space. Go to S3.2.6. Try another.

    3. Not this one. Animals compete for mates and shelter. Plants compete for light, water, space and nutrients. Go to S3.2.6. Try another.

    See the answer

    Light, water and space

    A plant needs light, water, space and nutrients. In a crowd there is less of each for every plant.

    • If you chose B: Plants make their own food from light. They compete for light, water and space. Go to S3.2.6.
    • If you chose C: Animals compete for mates and shelter. Plants compete for light, water, space and nutrients. Go to S3.2.6.

Keep it

Copy this into your exercise book.

Count plants in a few square frames put down at random. Find the mean count. Multiply by the number of frames that fit in the field. Random choice makes the sample fair.

Draw: Draw a field as a grid of squares. Mark five squares with frames and write a count in each.

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 (860 KB). It opens on any phone.