GCSE Maths: Sampling and population estimates (Foundation)
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Sampling and population estimates
The population is the whole group that you want to find out about, such as every student in a school or every fish in a lake. Asking everyone is called a census. It is often too slow, too expensive or impossible (you cannot test every battery, because testing destroys it). So we take a sample: a smaller group chosen from the population. We then use the sample to infer (work out) things about the population.
Random and biased samples. A sample is random if every member of the population has an equal chance of being chosen, for example by numbering everyone and using a random number generator or drawing names from a hat. A sample is biased if it unfairly favours some part of the population. Asking only the people leaving a gym about how often people exercise gives a biased result. A larger sample is usually more reliable, but only if it is chosen fairly: a large biased sample is still biased.
Estimating from a sample. If the sample is fair, the fraction of the sample with a property should be close to the fraction of the whole population. For example, if 14 out of a random sample of 50 students walk to school, then about 1450 of all the students walk. In a school of 1200, that is about 1450 × 1200 = 336 students. This is only an estimate. A different sample would give a slightly different answer.
Stratified sampling. If a population has clear groups (strata), such as year groups, a stratified sample takes the right number from each group so that the sample is a fair mini copy of the population. The number taken from each group is group size ÷ whole population × sample size. The method is fair because each group is represented in the same proportion as it appears in the population.
Capture–recapture. How can we count animals we cannot see? Catch a number of them, mark them harmlessly, release them, wait for them to mix back in, then catch a second sample. The fraction of the second sample that is marked should be about the same as the fraction of the whole population that is marked. So population estimate = (number marked first) × (second sample size) ÷ (number marked in the second sample).
This works only if some assumptions hold: the animals mix thoroughly, no animals are born, die or move in or out between the two catches, the marks do not fall off or harm the animals, and marked animals are no easier or harder to catch. If these are not true, the estimate may be too high or too low.
Key ideas
- Population = everyone or everything of interest. Sample = a smaller group chosen from it.
- A random sample gives every member an equal chance. A biased sample favours some members.
- Estimate from a sample: use the same fraction for the population.
- Stratified sample: group size ÷ population × sample size.
- Capture–recapture: population ≈ marked first × second sample ÷ marked in second sample.
- All estimates rely on assumptions and are not exact.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. A random sample should be like the whole population in miniature, so the same fraction of the population should have the property as in the sample.
- Write the fraction in the sample. Part over whole: the number with the property out of the sample size.1450 = 725
- Say that the same fraction applies to the whole population. This is the assumption that the sample is representative.725 of 1200
- Work out the fraction of the population. Multiply the fraction by the population size.0.28 × 1200 = 336
Check. The population is 24 times as big as the sample, and 14 × 24 = 336. ✓
Watch out. The answer is an estimate. A different sample would give a slightly different answer.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. Stratified sampling: take from each group (stratum) in proportion to its size.
- Find the size of the whole population. Add up the sizes of all the groups (strata).120 + 100 + 80 = 300
- Find the fraction of the population that is being sampled. In a stratified sample each group is sampled in the same proportion as its size in the population.30 out of 300: 30300 = 110
- Work out each group’s share of the sample. sample for the group = group size ÷ population size × sample size.Year 9: 120 ÷ 300 × 30 = 12
Year 10: 100 ÷ 300 × 30 = 10
Year 11: 80 ÷ 300 × 30 = 8 - Check the sample sizes add up to the sample. The groups together must make the whole sample.12 + 10 + 8 = 30
Check. Each group gets 110 of its size: 110 of 120 = 12, 110 of 100 = 10, 110 of 80 = 8. ✓
Watch out. Use the size of each group divided by the size of the whole population, not by the size of the sample.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. Capture–recapture: the fraction of the second sample that is marked is the same as the fraction of the whole population that is marked.
- Write down what we know. First we catch, mark and release some. Later we catch a second sample and count how many of those are marked.Marked and released: 40
Second sample: 30
Marked in the second sample: 6 - Write the proportion marked in the second sample. If the animals have mixed well, this is a fair estimate of the proportion marked in the whole population.6 out of 30 = 630
- Write the proportion marked in the whole population. Let N be the whole population. The 40 marked ones make up 40 out of N.40N
- Make the two proportions equal. We assume the second sample is representative of the population.630 = 40N
- Solve for N. Multiply both sides by N and by the second-sample size, then divide by the number marked in the second sample: N = marked × second sample ÷ marked in second sample.N = 40 × 30 ÷ 6 = 1200 ÷ 6 = 200
Check. Of the whole population of 200, 40 were marked: 40200 ≈ 0.200, and 630 = 0.200. The two proportions agree (to within rounding). ✓
Watch out. N = marked × second sample ÷ marked in the second sample. Do not divide by the second sample size or multiply by the number marked in it.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. Give every student an equal chance of being picked, and make the choice by chance, not by choice.
- Number the students. Give each student a different number from 1 to 28, perhaps using the register order.1, 2, 3, …, 28
- Choose numbers at random. Use a random number generator or draw numbers from a hat. Ignore repeats.Choose 5 different numbers between 1 and 28.
- Pick the students with those numbers. These students form the sample.Sample of 5 students
Check. Every student had the same chance, 5 out of 28, of being picked. ✓
Watch out. Choosing friends or the first five in the register is not random.
Watch out
For stratified sampling, divide by the size of the whole population and multiply by the sample size. And in capture–recapture, the denominator is the number marked in the second sample, not the first.Exercise 24B Sampling and population estimates
Non-calculatorKey words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. The population is everyone we want to find out about. The sample is the part we actually ask.
- Find the whole group of interest. The teacher wants to know about Year 10.Population = all 180 Year 10 students
- Find the group actually asked. Sample = the 20 students asked
Check. The sample (20) is smaller than the population (180). ✓
Watch out. The sample is always part of the population.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. The population is the whole batch, and the sample is the bulbs tested.
- Population. The factory wants to know about the whole batch.All 2000 bulbs
- Sample. The 50 bulbs tested
Check. 50 is much smaller than 2000. ✓
Watch out. Testing every bulb would use them all up, which is why a sample is used.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. Think about time, cost and what happens to the items.
- Cost and time. Asking or testing everybody can take a very long time and cost a lot.A sample is quicker and cheaper.
- Destructive testing. Some tests ruin the item (for example, testing how long a battery lasts).You cannot test every item if the test destroys it.
Check. Either reason is valid. ✓
Watch out. A sample is not used because it is more accurate: a census would be more accurate, but it is not practical.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. Decide whether everyone in the town had an equal chance of being asked.
- Think about who is at the bus station. People at a bus station are there because they use buses.They use the bus more than the typical person.
- Say what this does to the result. The sample is not representative of the whole town, so the results will overestimate bus use.The sample is biased.
Check. A better sample would be chosen at random from the whole town. ✓
Watch out. Say why the sample is unfair, not just that it is biased.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. A random sample gives every student the same chance of being chosen.
- Check A. Students who arrive early are not typical of everyone, so some have a better chance of being chosen.Not random.
- Check C. Only footballers can be chosen.Not random.
- Check B. A computer chooses from the full list, so every student has an equal chance.B
Check. Only B gives everyone the same chance. ✓
Watch out. “Random” means every member has an equal chance, not just that the choice looks haphazard.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. A random sample should be like the whole population in miniature, so the same fraction of the population should have the property as in the sample.
- Write the fraction in the sample. Part over whole: the number with the property out of the sample size.1040 = 14
- Say that the same fraction applies to the whole population. This is the assumption that the sample is representative.14 of 800
- Work out the fraction of the population. Multiply the fraction by the population size.0.25 × 800 = 200
Check. The population is 20 times as big as the sample, and 10 × 20 = 200. ✓
Watch out. The answer is an estimate. A different sample would give a slightly different answer.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. A random sample should be like the whole population in miniature, so the same fraction of the population should have the property as in the sample.
- Write the fraction in the sample. Part over whole: the number with the property out of the sample size.360 = 120
- Say that the same fraction applies to the whole population. This is the assumption that the sample is representative.120 of 3000
- Work out the fraction of the population. Multiply the fraction by the population size.0.05 × 3000 = 150
Check. The population is 50 times as big as the sample, and 3 × 50 = 150. ✓
Watch out. The answer is an estimate. A different sample would give a slightly different answer.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. A random sample should be like the whole population in miniature, so the same fraction of the population should have the property as in the sample.
- Write the fraction in the sample. Part over whole: the number with the property out of the sample size.925 = 925
- Say that the same fraction applies to the whole population. This is the assumption that the sample is representative.925 of 400
- Work out the fraction of the population. Multiply the fraction by the population size.0.36 × 400 = 144
Check. The population is 16 times as big as the sample, and 9 × 16 = 144. ✓
Watch out. The answer is an estimate. A different sample would give a slightly different answer.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. Capture–recapture: the fraction of the second sample that is marked is the same as the fraction of the whole population that is marked.
- Write down what we know. First we catch, mark and release some. Later we catch a second sample and count how many of those are marked.Marked and released: 30
Second sample: 40
Marked in the second sample: 6 - Write the proportion marked in the second sample. If the animals have mixed well, this is a fair estimate of the proportion marked in the whole population.6 out of 40 = 640
- Write the proportion marked in the whole population. Let N be the whole population. The 30 marked ones make up 30 out of N.30N
- Make the two proportions equal. We assume the second sample is representative of the population.640 = 30N
- Solve for N. Multiply both sides by N and by the second-sample size, then divide by the number marked in the second sample: N = marked × second sample ÷ marked in second sample.N = 30 × 40 ÷ 6 = 1200 ÷ 6 = 200
Check. Of the whole population of 200, 30 were marked: 30200 ≈ 0.150, and 640 = 0.150. The two proportions agree (to within rounding). ✓
Watch out. N = marked × second sample ÷ marked in the second sample. Do not divide by the second sample size or multiply by the number marked in it.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. Capture–recapture: the fraction of the second sample that is marked is the same as the fraction of the whole population that is marked.
- Write down what we know. First we catch, mark and release some. Later we catch a second sample and count how many of those are marked.Marked and released: 50
Second sample: 60
Marked in the second sample: 10 - Write the proportion marked in the second sample. If the animals have mixed well, this is a fair estimate of the proportion marked in the whole population.10 out of 60 = 1060
- Write the proportion marked in the whole population. Let N be the whole population. The 50 marked ones make up 50 out of N.50N
- Make the two proportions equal. We assume the second sample is representative of the population.1060 = 50N
- Solve for N. Multiply both sides by N and by the second-sample size, then divide by the number marked in the second sample: N = marked × second sample ÷ marked in second sample.N = 50 × 60 ÷ 10 = 3000 ÷ 10 = 300
Check. Of the whole population of 300, 50 were marked: 50300 ≈ 0.167, and 1060 = 0.167. The two proportions agree (to within rounding). ✓
Watch out. N = marked × second sample ÷ marked in the second sample. Do not divide by the second sample size or multiply by the number marked in it.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. Capture–recapture: the fraction of the second sample that is marked is the same as the fraction of the whole population that is marked.
- Write down what we know. First we catch, mark and release some. Later we catch a second sample and count how many of those are marked.Marked and released: 24
Second sample: 50
Marked in the second sample: 8 - Write the proportion marked in the second sample. If the animals have mixed well, this is a fair estimate of the proportion marked in the whole population.8 out of 50 = 850
- Write the proportion marked in the whole population. Let N be the whole population. The 24 marked ones make up 24 out of N.24N
- Make the two proportions equal. We assume the second sample is representative of the population.850 = 24N
- Solve for N. Multiply both sides by N and by the second-sample size, then divide by the number marked in the second sample: N = marked × second sample ÷ marked in second sample.N = 24 × 50 ÷ 8 = 1200 ÷ 8 = 150
Check. Of the whole population of 150, 24 were marked: 24150 ≈ 0.160, and 850 = 0.160. The two proportions agree (to within rounding). ✓
Watch out. N = marked × second sample ÷ marked in the second sample. Do not divide by the second sample size or multiply by the number marked in it.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. Capture–recapture: the fraction of the second sample that is marked is the same as the fraction of the whole population that is marked.
- Write down what we know. First we catch, mark and release some. Later we catch a second sample and count how many of those are marked.Marked and released: 120
Second sample: 150
Marked in the second sample: 25 - Write the proportion marked in the second sample. If the animals have mixed well, this is a fair estimate of the proportion marked in the whole population.25 out of 150 = 25150
- Write the proportion marked in the whole population. Let N be the whole population. The 120 marked ones make up 120 out of N.120N
- Make the two proportions equal. We assume the second sample is representative of the population.25150 = 120N
- Solve for N. Multiply both sides by N and by the second-sample size, then divide by the number marked in the second sample: N = marked × second sample ÷ marked in second sample.N = 120 × 150 ÷ 25 = 18000 ÷ 25 = 720
Check. Of the whole population of 720, 120 were marked: 120720 ≈ 0.167, and 25150 = 0.167. The two proportions agree (to within rounding). ✓
Watch out. N = marked × second sample ÷ marked in the second sample. Do not divide by the second sample size or multiply by the number marked in it.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. Think about what has to be true for the second sample to be a fair copy of the whole lake.
- Think about mixing. If the tagged fish stayed in one corner, the second sample would not be fair.The tagged fish have mixed evenly.
- Think about change. If fish were born, died or swam away in the week, the population changed.No fish were born, died or moved in or out.
Check. Any one of these is an acceptable answer. ✓
Watch out. State an assumption about the animals or the lake, not about the calculation.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. Stratified sampling: take from each group (stratum) in proportion to its size.
- Find the size of the whole population. Add up the sizes of all the groups (strata).90 + 60 + 150 = 300
- Find the fraction of the population that is being sampled. In a stratified sample each group is sampled in the same proportion as its size in the population.20 out of 300: 20300 = 115
- Work out each group’s share of the sample. sample for the group = group size ÷ population size × sample size.Year 7: 90 ÷ 300 × 20 = 6
Year 8: 60 ÷ 300 × 20 = 4
Year 9: 150 ÷ 300 × 20 = 10 - Check the sample sizes add up to the sample. The groups together must make the whole sample.6 + 4 + 10 = 20
Check. Each group gets 115 of its size: 115 of 90 = 6, 115 of 60 = 4, 115 of 150 = 10. ✓
Watch out. Use the size of each group divided by the size of the whole population, not by the size of the sample.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. Stratified sampling: take from each group (stratum) in proportion to its size.
- Find the size of the whole population. Add up the sizes of all the groups (strata).240 + 160 = 400
- Find the fraction of the population that is being sampled. In a stratified sample each group is sampled in the same proportion as its size in the population.50 out of 400: 50400 = 18
- Work out each group’s share of the sample. sample for the group = group size ÷ population size × sample size.Boys: 240 ÷ 400 × 50 = 30
Girls: 160 ÷ 400 × 50 = 20 - Check the sample sizes add up to the sample. The groups together must make the whole sample.30 + 20 = 50
Check. Each group gets 18 of its size: 18 of 240 = 30, 18 of 160 = 20. ✓
Watch out. Use the size of each group divided by the size of the whole population, not by the size of the sample.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. Stratified sampling: take from each group (stratum) in proportion to its size.
- Find the size of the whole population. Add up the sizes of all the groups (strata).125 + 100 + 75 = 300
- Find the fraction of the population that is being sampled. In a stratified sample each group is sampled in the same proportion as its size in the population.60 out of 300: 60300 = 15
- Work out each group’s share of the sample. sample for the group = group size ÷ population size × sample size.Year 10: 125 ÷ 300 × 60 = 25
Year 11: 100 ÷ 300 × 60 = 20
Sixth form: 75 ÷ 300 × 60 = 15 - Check the sample sizes add up to the sample. The groups together must make the whole sample.25 + 20 + 15 = 60
Check. Each group gets 15 of its size: 15 of 125 = 25, 15 of 100 = 20, 15 of 75 = 15. ✓
Watch out. Use the size of each group divided by the size of the whole population, not by the size of the sample.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. Stratified sampling: take from each group (stratum) in proportion to its size.
- Find the size of the whole population. Add up the sizes of all the groups (strata).124 + 96 + 80 = 300
- Find the fraction of the population that is being sampled. In a stratified sample each group is sampled in the same proportion as its size in the population.30 out of 300: 30300 = 110
- Work out each group’s share of the sample. sample for the group = group size ÷ population size × sample size.Juniors: 124 ÷ 300 × 30 = 12.40 = 12 (nearest whole number)
Seniors: 96 ÷ 300 × 30 = 9.60 = 10 (nearest whole number)
Veterans: 80 ÷ 300 × 30 = 8 - Check the sample sizes add up to the sample. The groups together must make the whole sample.12 + 10 + 8 = 30
Check. Each group gets 110 of its size: 110 of 124 = 12, 110 of 96 = 10, 110 of 80 = 8. ✓
Watch out. Use the size of each group divided by the size of the whole population, not by the size of the sample.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. Compare what a stratified sample should do with what Ben says.
- Recall the purpose. A stratified sample should be a fair small copy of the population, so groups must be represented in the same proportion as in the population.Bigger group, bigger share of the sample.
- Test Ben’s idea with an example. Suppose there are 90 students in one group and 10 in another. Taking 5 from each would over-represent the small group.Group of 90 and group of 10, sample of 10: take 9 and 1, not 5 and 5.
Check. 9 : 1 is the same ratio as 90 : 10. ✓
Watch out. Stratified means in proportion, not equal.
Key words. The population is everything we want to know about. A sample is a smaller part of it that we actually measure. A good sample is random (everyone has an equal chance) and representative (a fair mix).
Method. Think about what happens to the number of marked animals in the second sample.
- Consider the effect of marked animals dying. Fewer marked animals are in the lake, so fewer appear in the second sample.The number marked in the second sample, R, is smaller than it should be.
- Look at the formula. N = M × C ÷ R. If R is too small, then N is too big.The estimate is too high.
Check. Try M = 40, C = 30. With R = 6, N = 200. With R = 5, N = 240, which is larger. ✓
Watch out. A smaller R gives a bigger estimate.
Method. Use the formula N = M × C ÷ R twice, then compare the sizes of the samples.
- First estimate. Use M = 40, C = 30, R = 5.40 × 30 ÷ 5 = 1200 ÷ 5 = 240
- Second estimate. Use M = 40, C = 90, R = 14.40 × 90 ÷ 14 = 3600 ÷ 14 = 257.1 ≈ 257
- Choose which to trust. A larger sample gives a more reliable estimate, because it is less affected by chance.The second estimate (257) is more reliable because C = 90 is bigger than C = 30.
Check. Both estimates are close to each other (240 and 257), which suggests the population is about 250. ✓
Watch out. A bigger sample is more reliable, but only if it is fair.