How do we collect a sample that fairly represents a population?
Populations, samples and the census, sampling units and the sampling frame, and the main sampling methods including random, systematic, stratified, quota and opportunity sampling with their advantages and limitations.
A CCEA A-Level Mathematics answer on populations, samples and the census, the sampling frame and sampling units, and the main sampling methods - random, systematic, stratified, quota and opportunity - together with their advantages and limitations.
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What this dot point is asking
CCEA wants you to distinguish a population from a sample and a census, understand the sampling frame and sampling units, and describe the main sampling methods with their advantages and limitations. Good sampling underpins every statistical inference, so this dot point opens the statistics half of AS 2.
The answer
Population, sample and census
A census is accurate but expensive and often impractical; a sample is cheaper and faster, and is the only option when the population is huge or when testing destroys the item.
Random sampling methods
Stratified sampling
Non-random methods
In quota sampling the interviewer fills set numbers from each group but chooses individuals non-randomly, which is quick but can be biased. In opportunity (convenience) sampling you take whoever is available, which is easy but the least representative. Each method trades effort against the risk of bias.
Worked example: a systematic sample
Examples in context
Example 1. Quality control. A manufacturer testing the lifetime of light bulbs must use a sample, because testing is destructive: a census would destroy the entire stock. Sampling is not a compromise here, it is the only possibility.
Example 2. A school survey. To gauge opinion across year groups fairly, a stratified sample takes from each year in proportion to its size, so a large year is not drowned out by a small one. This is why stratification is preferred when the population has clear subgroups.
Try this
Q1. Define a census. [1 mark]
- Cue. A survey of every member of the population.
Q2. A population of is sampled with size by stratified sampling. A stratum has members. How many are sampled from it? [2 marks]
- Cue. .
Q3. Give one disadvantage of opportunity sampling. [1 mark]
- Cue. It is unlikely to be representative, so it can be biased.
Exam-style practice questions
Practice questions written in the style of CCEA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
CCEA 20215 marksA college has 600 Year 13 students, 360 of whom are female and 240 male. A sample of 50 is required. Describe how to take a stratified sample by gender, stating the number of each gender to include.Show worked answer →
In stratified sampling the sample reflects the proportions in the population. The sampling fraction is .
Female: .
Male: .
Then select that number from each stratum at random (for example, by numbering the members of each group and using random numbers).
So include females and males.
Markers reward the sampling fraction, the correct number in each stratum, and stating that selection within each stratum is random.
CCEA 20194 marksExplain the difference between a census and a sample, and give one advantage of each.Show worked answer →
A census collects data from every member of the population; a sample collects data from only a selected subset.
One advantage of a census is that it gives accurate, complete information about the whole population with no sampling error.
One advantage of a sample is that it is cheaper and quicker, and is the only option when testing is destructive or the population is very large.
Markers reward the definitions of census and sample, and a valid advantage of each.
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Sources & how we know this
- CCEA GCE Mathematics specification — CCEA (2018)