Statistics: Sampling
Sampling is the process of selecting a smaller group, or sample, from a larger population so that data can be collected and conclusions drawn about the whole population without surveying everyone. A Level Statistics covers the difference between a census and a sample, sampling frames, and named sampling methods including simple random, systematic, stratified, quota and opportunity sampling.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Decide whether a census (every member of the population) or a sample (part of the population) is more appropriate, weighing accuracy against time, cost and practicality.
- Identify the sampling frame: the list of all members of the population from which the sample will be drawn, and check whether it is complete and up to date.
- For a simple random sample, assign each member of the frame a unique number and use a random number generator (or random number table) to select the required number, rejecting any numbers outside the range or already chosen.
- For a systematic sample of size n from a population of size N, calculate the sampling interval k = N/n (rounded down), choose a random starting point between 1 and k, then select every kth member after that.
- For a stratified sample, divide the population into strata (groups), then select from each stratum a number proportional to its size in the population: (stratum size / population size) x sample size.
- State the advantages and disadvantages of the method used, referring to bias, cost, time and whether every member has a known (or equal) chance of selection.
Worked example
A college has 720 students: 300 in Year 12 and 420 in Year 13. The college wants a stratified sample of 60 students, by year group. (a) Find the number of Year 12 students needed in the sample. (b) Find the number of Year 13 students needed in the sample. (c) Describe how the college should select which Year 12 students to include.
- Fraction of the population in Year 12 = 300/720.
- Number of Year 12 students in the sample = 60 x (300/720) = 25.
- Number of Year 13 students in the sample = 60 x (420/720) = 35 (or 60 - 25 = 35).
- To select which Year 12 students to include, number all 300 Year 12 students and use simple random sampling (e.g. a random number generator) to choose 25 of them.
Practice questions
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Q1State what is meant by a census.Show answer
Answer: A survey that collects data from every member of the population.
Q2A school has an up to date, numbered list of all 1200 students. State the name for this list, in the context of sampling.Show answer
Answer: The sampling frame.
Q3A researcher wants a systematic sample of 20 from a population of 340. Calculate the sampling interval, k.Show answer
Answer: k = 17 (340 divided by 20).
Q4A gym has 450 members: 180 are under 30 and 270 are 30 or over. Find the number of under-30 members needed for a stratified sample of size 50.Show answer
Answer: 20 (50 x 180/450).
Q5A radio show asks listeners to call in and give their opinion on a new policy. State the name of this sampling method, and give one way it could produce a biased sample.Show answer
Answer: Self-selecting (volunteer) sampling; only listeners who feel strongly enough to call in respond, so the sample may over-represent extreme opinions.
Q6A stable has 60 horses, numbered 00 to 59. Describe fully how a simple random sample of 5 horses could be selected using a random number generator, ensuring no horse is chosen twice.Show answer
Answer: Generate two-digit random numbers, reject any greater than 59 and any repeats, and continue until 5 different horses have been selected.
Exam-style questions
Written in the style of a A Level Maths exam paper, with a full mark scheme.
A company holds a complete computer record of the annual mileage driven by each of its 3000 company cars. (a) State what is meant by a census. (1) (b) Give one advantage of the company using a sample of 150 cars rather than a census. (1) (c) Give one disadvantage of using a sample rather than a census in this case. (2)
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A supermarket had 640 customers pass through its tills last Saturday morning. The manager wants to select a systematic sample of 20 of these customers to ask about checkout waiting times. (a) Calculate the sampling interval, k, the manager should use. (2) (b) Describe fully how the manager should select the systematic sample of 20 customers. (2) (c) The manager considers only sampling customers between 9am and 10am to make the sample quicker to select. Explain why this could introduce bias. (1)
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A large secondary school has 1400 students across Years 7 to 13, and the school office holds a complete, up to date list of every student with their year group. (a) Give one reason why taking a sample, rather than a census of all 1400 students, would be appropriate here. (1) (b) Recommend a suitable sampling method for the head teacher to use, and justify your choice with reference to the information given. (3) (c) State one limitation of the method you recommended in part (b). (2)
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Free printable worksheet
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