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A sample is part of a population. Learn how to select a random sample, explain bias, and use the sample proportion to estimate a number in the whole population.
Key facts
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
A larger random sample gives a more reliable estimate
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.
How to answer each type of question
M = method mark, A = answer mark, B = a mark on its own.
Explain why a sample may be biased and how to improve it
1 to 2 marksGrade 3
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.
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(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 marksGrade 4
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.
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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 marksGrade 4
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.
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\(\dfrac{4}{150} \times 6000\) (M1) = 160 (A1)
Don’t lose marks
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. \(\dfrac{4}{150}\)) 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.
More tips
Memory tricks
Bias test: could everyone in the population have been picked with the same chance? If not, the sample may be biased.
Write the sample result as a fraction before scaling it to the population.
Exam technique
Answer sampling questions in context: name the people, places or times in the question.
An estimate is uncertain because a sample may not match the whole population exactly.
What each grade needs
What you need to be able to do, from the first marks up to the top grade.
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.
Grade 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.
Grade 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.
Grade 4
Use a sample to estimate for a populationScale up the proportion in the sample: proportion × population size.
Grade 4
Suggest how to make a sample more reliableUse a larger random sample, taken from the whole population rather than one place or time.
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
There are 900 students at a school. Kai asks a random sample of 25 of the students how many books they read last month. The table shows information about Kai’s results.
[object Object]
(a) Work out an estimate for the total number of books read last month by all 900 students.[3]
(b) Explain why it is important that Kai’s sample was chosen at random.[1]
Show the answer and mark scheme
(a)Answer: 1080
M1 for \(\Sigma fx = 30\) (at least three correct products, e.g. \(1 \times 10\))
M1 for \(\frac{30}{25} \times 900\) oe
A1 for 1080
Worked solution: \(\Sigma fx = 0 \times 8 + 1 \times 10 + 2 \times 4 + 3 \times 1 + 4 \times 1 + 5 \times 1 = 30\). The 25 students in the sample read 30 books altogether. Estimate \(= \frac{30}{25} \times 900 = 1080\).
(b)Answer: So that the sample is not biased and is likely to be representative of all 900 students.
C1 for a correct explanation, e.g. so that the sample is representative / not biased / every student has the same chance of being chosen