Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Statistics
Practise Sampling and capture–recapture. 1 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Populations and samples: why samples are used, how to take a random sample, why a sample may be biased, how to use a sample to estimate for a whole population, and capture–recapture to estimate the size of an animal population. This comes up on both tiers, usually as short 'explain' questions or a capture–recapture calculation.
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
2
Know the words population, sample and censusThe population is the whole group; a sample is part of it; a census collects data from every member.
3
Explain why a sample may be biasedSay which group is more or less likely to be chosen, so the sample does not represent the population.
4
Describe how to take a random sampleNumber every member of the population and use random numbers to choose, so each has an equal chance.
4
Use a sample to estimate for a populationScale up the proportion in the sample: proportion × population size.
4
Suggest how to make a sample more reliableUse a larger random sample, taken from the whole population rather than one place or time.
5
Estimate a population using capture–recaptureAssume the proportion marked in the second sample equals the proportion marked in the whole population.
Notes
Populations and samples
The population is the whole group you want to find out about, e.g. all 900 students at a school.
A census collects data from every member of the population. A sample collects data from only part of it.
A sample is quicker and cheaper, and is the only option when testing destroys the item (e.g. testing how long batteries last).
A sample may not represent the population, so any conclusion is only an estimate. A larger sample usually gives a more reliable estimate.
Random samples and bias
In a random sample, every member of the population has an equal chance of being chosen.
To take one, list the population, give each member a number, then use a random number generator (or a calculator) to choose numbers, ignoring repeats.
A biased sample does not represent the population fairly, because some groups are more likely to be chosen than others, e.g. asking only people at a gym how often people exercise.
A good sample includes the different groups (ages, places, times of day) in roughly the same proportions as the population.
Estimating from a sample
Estimate for the population = (number in the sample with the feature ÷ sample size) × population size.
e.g. 12 out of a random sample of 50 students walk to school. For a school of 900 students: 12 ÷ 50 × 900 = 216 students.
Capture–recapture
Catch M animals, mark them and release them.
Later, catch a second sample of C animals and count how many of them, R, are marked.
Assume the fraction marked is the same in the second sample as in the whole population, N: \(\frac{R}{C} = \frac{M}{N}\), so \(N = \frac{M \times C}{R}\).
Cheatsheet
Population: the whole group being studied
Census: data from every member; sample: data from part of the population
Random sample: every member has an equal chance of being chosen
Biased sample: some groups are more likely to be chosen, so it does not represent the population
Estimate for the population = proportion in the sample × population size
Capture–recapture: \(N = \frac{M \times C}{R}\)
M = number marked the first time, C = number caught the second time, R = number of these that are marked
A larger random sample gives a more reliable estimate
How to answer each type of question
Explain why a sample may be biased and how to improve it
1 to 2 marks3
Ask: could every member of the population have been chosen with the same chance?
Name the group that is over- or under-represented, in context.
To improve it, suggest a random sample from the whole population, or a larger one.
Example. Amir wants to find out how often the adults in his town use the leisure centre. He asks 40 adults as they leave the leisure centre on a Saturday morning. (a) Give one reason why his sample is likely to be biased. (b) Suggest one way he could improve his sample.
Show the model answer
(a) Everyone he asks has just used the leisure centre, so they are likely to use it more often than other adults in the town (C1) (b) Choose a random sample of adults from the whole town, e.g. from a numbered list using random numbers (C1)
Describe how to take a random sample
2 marks4
Say how you get a list of the whole population and number it.
Say how you choose: random numbers from a calculator or computer, ignoring repeats.
Say how many you choose.
Example. A school has 850 students. Describe how the head teacher could choose a random sample of 50 of these students.
Show the model answer
Use the school register to give each student a different number from 1 to 850 (C1) Use a random number generator to choose 50 different numbers, ignoring repeats, and select those students (C1)
Use a sample to estimate for the population
2 marks4
Write the result of the sample as a fraction of the sample size.
Multiply by the size of the population.
Round to a whole number if you are counting items.
Example. A factory makes 6000 light bulbs each day. A random sample of 150 of these bulbs contains 4 faulty bulbs. Work out an estimate for the number of faulty bulbs the factory makes each day.
Show the model answer
\(\frac{4}{150} \times 6000\) (M1) = 160 (A1)
Estimate a population using capture–recapture
3 marks5
Identify M (marked the first time), C (caught the second time) and R (marked in the second sample).
Write the proportion equation \(\frac{R}{C} = \frac{M}{N}\).
Solve it: \(N = \frac{M \times C}{R}\), and give a whole number.
Example. Priya wants to estimate the number of fish in a pond. She catches 40 fish, marks each one and puts them back in the pond. A week later she catches 50 fish. 8 of these fish are marked. Work out an estimate for the number of fish in the pond.
Show the model answer
\(\frac{8}{50} = \frac{40}{N}\), or 50 ÷ 8 = 6.25 (M1) \(N = \frac{40 \times 50}{8}\) (M1) = 250 fish (A1)
Shortcuts and memory tricks
Capture–recapture as two equal fractions: marked in the second sample ÷ size of the second sample = number marked ÷ population.
'M times C over R': multiply the two catches, then divide by the number of marked animals recaptured.
Sense check: a population estimate can never be smaller than either sample.
Bias test: could everyone in the population have been picked with the same chance? If not, the sample may be biased.
Where marks are lost
Dividing by the wrong number in capture–recapture. The bottom of \(\frac{M \times C}{R}\) is the number of marked animals in the second sample.
Writing 'pick people randomly' without saying how. Describe numbering the population and choosing with random numbers.
Saying only 'the sample is too small' when asked why it is biased. A small sample is less reliable, but bias means some groups are more likely to be chosen, so name the group.
Giving the proportion in the sample (e.g. \(\frac{4}{150}\)) instead of scaling it up to the population.
Leaving a population estimate as a decimal, such as 312.5 animals. Round it to a whole number.
Exam technique
Answer sampling questions in context: name the people, places or times in the question.
In capture–recapture, write the fractions or the formula before you substitute. This earns the first method mark.
When asked if an estimate is too high or too low, state the direction first, then give the reason linked to the number of marked animals recaptured.
For 'describe how to take a random sample', give two steps: numbering the whole population, and choosing with random numbers.
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy2 marks
Amir wants to find out how much time adults in a city spend on the internet. Amir puts a survey on a website and uses the first 100 replies.
(a) Give one reason why this sample may be biased.[1]
(b) Suggest one way in which Amir could improve the sample.[1]
Show the answer and mark scheme
(a)Answer: Only people who use the internet (and choose to reply) are included.
C1 for a correct reason, e.g. only people who use the internet (and choose to reply) are included
(b)Answer: Choose adults at random from the whole city and survey them in a way that does not need the internet.
C1 for a sensible improvement, e.g. choose adults at random from the whole city and survey them in a way that does not need the internet
Question 2Medium4 marks
Ruth wants to estimate the number of fish in a pond. She catches 40 fish, puts a tag on each one and returns them to the pond. A week later she catches 50 fish. 8 of these fish have a tag.
(a) Work out an estimate for the number of fish in the pond.[2]
(b) State one assumption that Ruth has made.[1]
(c) Some of the tags fell off before the second catch. Explain what effect this has on Ruth's estimate.[1]
Show the answer and mark scheme
(a)Answer: 250
M1 for \(\frac{40 \times 50}{8}\) or \(\frac{8}{50} = \frac{40}{N}\)
A1 for 250
Worked solution: \(\frac{8}{50} = \frac{40}{N}\), so \(N = \frac{40 \times 50}{8} = 250\).
(b)Answer: E.g. the number of fish in the pond does not change during the week.
B1 for a correct assumption, e.g. no fish are born, die, arrive or leave; the tagged fish mix evenly with the others; tags do not fall off; every fish is equally likely to be caught
Worked solution: The method assumes the population is the same size at both catches and the tagged fish are spread evenly through it.
(c)Answer: Fewer tagged fish are caught than should be, so 8 is too small and the estimate is too large (an overestimate).
C1 for explaining that fewer tagged fish are recaught, so the estimate is too high
Worked solution: Dividing by a number that is too small (fewer tagged fish recaught) makes \(\frac{40 \times 50}{8}\) too large.