Notes · MathematicsUK · A-Levels

Statistical sampling

Statistical sampling is the study of how a sample is drawn from a population so that conclusions about the whole can be made from a part. The A-Level covers populations, censuses and samples, the main random sampling methods (simple random, systematic, stratified) and non-random methods (opportunity, quota), their advantages and drawbacks, and sampling in the context of the prescribed large data set.

4 sections~11 min reading time3 competenciesLevel Foundation 1 · Standard 3

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Exam profile

AO1 · Describe and carry out standard sampling methodsAO2 · Evaluate sampling methods and the bias they may introduceAO3 · Relate sampling to the large data set and real contexts

Operators:describeexplainstateevaluatesuggest

basic level

AS-Level covers populations and samples and the basic sampling methods with their advantages and disadvantages.

higher level

The full A-Level expects fluent evaluation of methods and their application to the large data set.

Depth

Text

Contents · 4 sections

Statistical sampling

Populations, censuses and samples#

FoundationDfE GCE Mathematics — samplingAQA 7357 (Statistics)

From population to inference

Fig. 1A statistic computed from the sample is used to infer a parameter of the whole population.

A population is the entire collection of items or people under investigation, and a sample is a selected subset of it. The individual members are sometimes called sampling units, and a list of all of them is a sampling frame. Studying a sample rather than the whole population is the central compromise of statistics: you accept some uncertainty in return for a study that is actually feasible.

A census collects data from every member of the population, while a sample survey collects from a subset. A census gives a complete and unbiased picture but is often expensive, time-consuming, and sometimes impossible — testing every battery to destruction, for instance, would leave none to sell. A sample is cheaper and quicker and is usually the only practical option, at the cost of some sampling error.

The purpose of sampling is inference: you calculate a statistic from the sample (such as the sample mean

\bar{x}

) and use it to estimate the corresponding population parameter (such as the population mean

\mu

). A good sampling method makes the sample representative of the population, so that the statistic is a reliable estimate; a poor method introduces bias, a systematic tendency to over- or under-estimate.

The key trade-offs to be able to discuss are cost and time (favouring a sample) against completeness and accuracy (favouring a census), and the crucial requirement that a sample be representative. A larger sample generally gives a more reliable estimate, but only if the method of selection is itself sound — a huge biased sample is worse than a small fair one.

Worked example

Census or sample?

A supermarket wishes to know the average weight of the apples in a delivery of 20 000 apples. Discuss whether to use a census or a sample.

Consider a census
Weighing all 20 000 apples would be accurate but very time-consuming and costly.
Consider a sample
Weighing, say, 100 apples selected fairly gives a quick estimate of the mean at much lower cost.
Decide
A sample is appropriate here: apples are similar enough that a representative sample estimates the mean well, and no apples are destroyed by weighing.

Result: A sample survey is the sensible choice, balancing accuracy against cost and time.

Typical mistakes

Confusing the population with the sample, or a parameter with a statistic.
Assuming a census is always best, ignoring cost, time and destructive-testing constraints.

Active revision

A factory wants to know the mean lifetime of the light bulbs it produces. Explain why a census is not appropriate and state what the population and a suitable sample would be.

Active recall

Recall the key points — then reveal.

Sources: Mathematics: AS and A level content (GCE subject content) (Department for Education) · AQA A-level Mathematics 7357 specification (AQA)

Random sampling methods#

StandardDfE GCE Mathematics — samplingAQA 7357 (Statistics)

A proportional stratified sample

Fig. 2A sample of 50 from a school of 1000 (year groups 400, 300, 200, 100) allocates

20, 15, 10, 5

proportionally.

In a simple random sample of size

n

, every possible sample of that size is equally likely to be chosen; equivalently, each member of the population has the same chance of selection. In practice you number the population using the sampling frame and choose

n

different numbers using random numbers or a random-number generator. It is unbiased and simple to justify, but it requires a full sampling frame and can be cumbersome for a large population.

Systematic sampling selects members at regular intervals from an ordered list: choose a random starting point among the first

k

and then take every

k

th member, where

k = \dfrac{\text{population size}}{\text{sample size}}

. It is quick and spreads the sample through the list, but it can introduce bias if there is a periodic pattern in the list that coincides with the interval.

Stratified sampling divides the population into distinct groups (strata) — such as age bands, school years or genders — and samples from each stratum in proportion to its size. The number taken from a stratum is

\dfrac{\text{stratum size}}{\text{population size}} \times \text{sample size}

, and within each stratum a simple random sample is taken. It guarantees that each group is represented in the right proportion, giving a very representative sample when the strata are relevant to the variable studied.

Each random method has a place: simple random sampling is the fairest baseline, systematic sampling is efficient for ordered lists, and stratified sampling is best when the population contains clearly different groups. Being able to describe precisely how to carry each one out — not merely name it — is what the examiners look for.

\text{stratum sample} = \frac{\text{stratum size}}{\text{population size}} \times \text{sample size}

Proportional stratified allocation

Each stratum contributes a number proportional to its share of the population.

Worked example

A stratified allocation

A company has 240 managers, 360 office staff and 600 factory workers. Find the numbers in a stratified sample of size 100.

Total population
$240 + 360 + 600 = 1200$ .
Sampling fraction
$\dfrac{100}{1200} = \dfrac{1}{12}$ .
Allocate proportionally
Managers $\tfrac{240}{12} = 20$ ; office $\tfrac{360}{12} = 30$ ; factory $\tfrac{600}{12} = 50$ .
$20 + 30 + 50 = 100$

Result: 20 managers, 30 office staff and 50 factory workers.

Typical mistakes

Naming a method without describing how it is actually carried out.
Allocating a stratified sample equally rather than in proportion to stratum sizes.

Active revision

A college of 1500 students has 600 in Year 12 and 900 in Year 13. Describe how to take a stratified sample of 60 students, stating how many come from each year.

Active recall

Recall the key points — then reveal.

Sources: Mathematics: AS and A level content (GCE subject content) (Department for Education) · AQA A-level Mathematics 7357 specification (AQA)

Non-random sampling methods#

StandardDfE GCE Mathematics — samplingAQA 7357 (Statistics)

Comparison of sampling methods

Fig. 3Random methods support statistical inference; non-random methods trade rigour for speed.

Non-random methods do not give every member a known chance of selection, so they are quicker but risk bias. Opportunity sampling (also called convenience sampling) takes whoever is available at the time — the first 20 people who pass a survey point, say. It is very easy and cheap, but is often unrepresentative because the people available may differ systematically from the wider population.

Quota sampling instructs the interviewer to select a fixed number of people from each of several categories (for example 10 men and 10 women), filling each quota by opportunity. It is used widely in market research because it is fast and captures the required group proportions, but because the final selection within each quota is not random it can still be biased by the interviewer's choices.

The essential contrast is that random methods allow the theory of probability to quantify the reliability of an estimate, whereas non-random methods do not. This is why hypothesis testing and confidence statements assume random sampling; a conclusion drawn from an opportunity sample cannot claim the same statistical support.

In an exam you should be able to identify which method has been used, describe how to carry it out, and — most importantly — evaluate it, naming a specific source of bias in the given context. 'The first 30 shoppers may all be there at a quiet time and not represent weekend customers' is the kind of context-specific criticism that earns the AO2 marks.

Worked example

Evaluating a sampling method

A student surveys classmates in her own maths class about how much homework students in the school get. Identify the method and evaluate it.

Identify the method
She samples those conveniently available — an opportunity sample.
Identify bias
Her maths class is not representative: they may take similar subjects and get similar homework, differing from other students.
Suggest an improvement
A stratified sample across year groups and subject choices would better represent the whole school.

Result: It is an opportunity sample, likely biased; a stratified sample across the school would be more representative.

Typical mistakes

Giving a generic criticism ('it may be biased') instead of one rooted in the context.
Confusing quota sampling (non-random selection within categories) with stratified sampling (random within strata).

Active revision

A researcher interviews the first 25 people leaving a gym on a Monday morning about their exercise habits. Name the sampling method and give two reasons why the sample may not be representative of all gym members.

Active recall

Recall the key points — then reveal.

Sources: Mathematics: AS and A level content (GCE subject content) (Department for Education) · AQA A-level Mathematics 7357 specification (AQA)

Sampling and the large data set#

StandardDfE GCE Mathematics — samplingAQA 7357 (Statistics)

Types of variable in a data set

Fig. 4Knowing the type of each variable guides which diagrams and statistics are appropriate.

Every A-Level Mathematics specification prescribes a large data set that students study during the course, and examination questions in statistics are often set in its context. Familiarity with the data set — the variables it records, their units, and the kinds of values they take — means you can interpret a question quickly and spot values that are implausible or clearly errors.

The large data set is itself usually a sample or a collection of records drawn from a much larger population, so the ideas of this topic apply directly to it. You should be able to describe how a sample could be drawn from the data set, for instance by numbering the rows and taking a simple random or systematic sample, and to discuss whether that sample is representative of the population the data set came from.

Working with real data introduces practical issues absent from tidy textbook numbers: missing entries, obvious recording errors, and variables measured on different scales or in different units. Recognising and handling these — for example by discarding a plainly erroneous value with justification — is part of the statistical literacy the specification is developing.

The data set also connects to the later statistics topics: its variables provide realistic contexts for calculating measures of location and spread, drawing diagrams, examining correlation, and testing hypotheses. Treating it as a familiar object, rather than as an unseen table, is a genuine advantage in the Paper 3 statistics questions.

Worked example

Sampling from a data set

A data set has 500 records numbered 1 to 500. Describe how to take a systematic sample of 25 records.

Find the interval
$k = \dfrac{500}{25} = 20$ , so take every 20th record.
Random start
Choose a random starting number between 1 and 20, say 7.
Select
Take records 7, 27, 47, 67, ... up to 487 — 25 records in all.

Result: Every 20th record from a random start (e.g. 7, 27, 47, ...) gives a systematic sample of 25.

Typical mistakes

Treating the data set as a full population when it is itself a sample of a larger one.
Ignoring units or misreading a variable's scale when interpreting values.

Active revision

Describe how you would use random numbers to select a simple random sample of 15 rows from a data set of 300 records, and state one limitation of the resulting sample.

Active recall

Recall the key points — then reveal.

Sources: Mathematics: AS and A level content (GCE subject content) (Department for Education) · AQA A-level Mathematics 7357 specification (AQA)

References & sources

Sources

Department for Education

Mathematics: AS and A level content (GCE subject content)

AQA

AQA A-level Mathematics 7357 specification

Statistical sampling

4 sections~11 min reading time3 competenciesLevel Foundation 1 · Standard 3

Statistical sampling

Key points

Census or sample?

Key points

A stratified allocation

Key points

Evaluating a sampling method

Key points

Sampling from a data set

Sources

Statistical sampling

Key points

Census or sample?

Key points

A stratified allocation

Key points

Evaluating a sampling method

Key points

Sampling from a data set

Sources