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How do you select a representative sample from a population, and what are the trade-offs of each method?

Populations and samples, census and sampling, random and non-random sampling methods, and the advantages and limitations of each in context.

A focused answer to the Edexcel A-Level Mathematics statistical sampling content, covering populations and samples, the difference between a census and a sample, random and non-random sampling methods, and the advantages and limitations of each in context.

Generated by Claude Opus 4.88 min answer

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What this dot point is asking

Edexcel wants you to understand the terms population and sample, distinguish a census from a sample, describe and evaluate the main sampling methods (simple random, systematic, stratified, quota and opportunity sampling), and explain the advantages and limitations of each, including in the context of the large data set.

The answer

Census versus sample

A census collects data from every member of the population. It is accurate and avoids sampling error, but is expensive, slow and sometimes impossible (for example when testing destroys the item). A sample is quicker and cheaper, but only estimates population features and carries sampling error.

Random sampling methods

Systematic sampling takes every kkth member from an ordered list after a random start, where k=population sizesample sizek = \dfrac{\text{population size}}{\text{sample size}}. It is quick to administer from an ordered list, but a periodic pattern in the list at the same interval as kk can bias the sample. Stratified sampling divides the population into groups (strata) and samples each in proportion to its size, which represents subgroups well; the size taken from each stratum is the stratum's size multiplied by the overall sampling fraction.

Non-random sampling methods

Quota sampling fills set numbers from each group without random selection, and opportunity (convenience) sampling uses whoever is available. Both are quick and cheap but can be biased and do not support probability statements. The key distinction from stratified sampling is that quota sampling does not select randomly within each group, so a confident probability statement about the population cannot be justified.

Examples in context

Try this

Q1. A school of 800 students has 480 girls and 320 boys. Describe how to take a stratified sample of 50. [3 marks]

  • Cue. Sample 480800×50=30\tfrac{480}{800} \times 50 = 30 girls and 320800×50=20\tfrac{320}{800} \times 50 = 20 boys, chosen at random within each group.

Q2. Give one advantage and one disadvantage of an opportunity sample. [2 marks]

  • Cue. Advantage: quick and cheap. Disadvantage: likely biased and not representative.

Exam-style practice questions

Practice questions written in the style of Pearson Edexcel exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Edexcel 20185 marksA company has 12001200 employees: 720720 in production, 300300 in sales and 180180 in administration. A stratified sample of size 4040 is required. Calculate the number sampled from each department and describe how the individuals within a department are selected.
Show worked answer →

The sampling fraction is 401200=130\dfrac{40}{1200} = \dfrac{1}{30} (M1).

Production: 72030=24\dfrac{720}{30} = 24; sales: 30030=10\dfrac{300}{30} = 10; administration: 18030=6\dfrac{180}{30} = 6 (A1, A1). These total 4040.

Within each department, number the employees and use random numbers to select the required count, so each has an equal chance (M1, A1).

Markers reward the sampling fraction, the three correct strata sizes summing to 4040, and a valid random selection within strata.

Edexcel 20214 marksA factory produces 20002000 items in a shift, listed in production order. Describe how to take a systematic sample of 5050 items, and state one advantage and one disadvantage of systematic over simple random sampling.
Show worked answer →

The sampling interval is k=200050=40k = \dfrac{2000}{50} = 40 (M1, A1).

Choose a random starting number between 11 and 4040, then select every 4040th item thereafter (M1).

Advantage: it is quick and easy to administer from an ordered list. Disadvantage: a periodic pattern in the list at the interval of 4040 could bias the sample (A1).

Markers reward the interval, the random start with the every-kkth rule, and a valid advantage and disadvantage.

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