K1StatisticsFoundation

Sampling Methods

You almost never get to measure everyone. Statistics is the business of saying something reliable about a whole $\textbf{population}$ from a much smaller $\textbf{sample}$ — and whether that works depends entirely on how the sample was chosen.

30 min Statistical Sampling

The big picture

Every method below trades representativeness against practicality, and no method wins on both. A simple random sample is unbiased but needs a complete list of the population and a lot of legwork; an opportunity sample takes ten minutes and can be badly skewed. The examinable skill is not memorising five definitions — it is being able to say, for a given situation, which method is appropriate and what its specific weakness is. That judgement is also what makes the rest of statistics meaningful: every hypothesis test later assumes the sample was drawn properly, and none of the mathematics rescues you from a biased sample.

What you'll be able to do

Define population, sample, sampling frame and census, and give advantages of each
Describe simple random, systematic, stratified, quota and opportunity sampling
Select an appropriate method for a given context and justify it
Identify sources of bias in a described sampling procedure

Census or sample?

A $\textbf{census}$ observes every member of the population. A $\textbf{sample}$ observes a subset. The $\textbf{sampling frame}$ is the list of population members from which the sample is drawn — a school register, an employee database, an electoral roll.

A census gives a completely accurate picture, but it is expensive, slow, and sometimes impossible: testing every match in a factory’s output would leave nothing to sell, because the test destroys the item. A sample is cheaper and faster but only ever an estimate, and it can be biased.

Not every population has a usable sampling frame. You cannot list every fish in a lake, which immediately rules out any method that requires numbering the population.

Why this works

Destructive testing is the cleanest argument for sampling. If checking a fuse means blowing it, a census would destroy the entire product line — so the question is never "sample or census?" but "how do we sample well?".

Random methods

In a $\textbf{simple random sample}$ , every member of the population has an equal chance of selection and every possible sample is equally likely. In practice you number the sampling frame and generate random numbers, discarding repeats. It is unbiased, but needs a full frame and can be laborious.

$\textbf{Systematic sampling}$ takes every $k$ th member from an ordered list, where $k$ is the population size divided by the sample size, with a random start in the first block. It is quicker than simple random sampling and spreads the sample through the list — but if the list has a repeating pattern with the same period as $k$ , the sample can be badly skewed.

$\textbf{Stratified sampling}$ divides the population into groups (strata) that matter — year groups, departments, age bands — and samples from each in proportion to its size. It guarantees every group is represented in the right ratio, which simple random sampling only does on average.

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

Systematic sampling interval.

n_{\text{stratum}} = \dfrac{\text{stratum size}}{\text{population size}} \times n

Stratified sample size for each group.

Example 1A school of 1200 students has 480 in Key Stage 3, 420 in Key Stage 4 and 300 in the sixth form. Find the stratified sample sizes for a sample of 80.

1Sampling fraction:

\dfrac{80}{1200} = \dfrac{1}{15}

2KS3:

\dfrac{480}{15} = 32

3KS4:

\dfrac{420}{15} = 28

4Sixth form:

\dfrac{300}{15} = 20

Answer32, 28 and 20 — which total 80 as required.

Tip — Always check your stratified sample sizes add to the total. If rounding has left you one over or under, adjust the largest stratum and say that you have done so.

Non-random methods

$\textbf{Quota sampling}$ sets targets for how many of each type to interview — 20 men and 20 women, say — and the interviewer chooses who fills the quota. It is cheap and needs no sampling frame, but the interviewer’s choices introduce bias: people who look approachable get picked disproportionately.

$\textbf{Opportunity sampling}$ (also called convenience sampling) takes whoever is available at the time. It is the fastest method and the least reliable, because the people who happen to be in one place at one time are rarely typical of the population.

Neither method is random, so strictly the inference machinery of later chapters does not apply. They appear in the specification because they are common in real research and you need to be able to criticise them.

Example 1A researcher surveys shoppers outside a supermarket at 10am on a Tuesday about commuting habits. Identify two sources of bias.

1The time excludes anyone at work — precisely the commuters the study is about.

2The location captures only people who shop at that supermarket, not the wider population.

AnswerTime-of-day bias excluding working commuters, and location bias restricting the sample to one shop’s customers.

Why this works

The strongest criticism of a non-random sample is usually specific rather than general. "It is biased" earns little; "surveying at 10am on a weekday systematically excludes people in full-time work, who are the population of interest" earns the marks.

Choosing and justifying a method

Exam questions give a scenario and ask which method is appropriate, or ask you to criticise one. Two questions settle most cases: is there a usable sampling frame, and does the population contain groups that must be represented proportionally?

No frame available points to quota or opportunity sampling. A frame plus important subgroups points to stratified. A frame with no meaningful subgroups allows simple random or systematic.

Whatever you choose, name the specific advantage and the specific drawback in context rather than in the abstract.

Example 1A factory manager wants to check the quality of items on a production line. Suggest a suitable sampling method with a reason.

1Items come off the line in a natural order, so a list exists implicitly.

2Systematic sampling — take every 50th item — is easy to implement on a moving line.

3The caution: if a fault recurs with the same period as the interval, it would be missed or over-counted.

AnswerSystematic sampling, because it is simple to operate on a production line and spreads the sample across the whole run.

Tip — When asked for an advantage, make it about this scenario. "Quick and easy" is weak; "no sampling frame is needed, and the shoppers are already present" is a reason.

Think like an examiner

Define terms precisely: a simple random sample gives every $\textit{sample}$ an equal chance, not merely every individual.
Show the sampling-fraction working in stratified questions and confirm the parts sum to the total.
Criticisms must be contextual. Name who is excluded or over-represented and why.
If a method needs a sampling frame and the question says none exists, say so — that is usually the mark.

Common misconceptions

Calling any convenient sample "random".

Random requires a defined chance mechanism. Choosing whoever is nearby is opportunity sampling.

Rounding stratified sample sizes independently and ending up with the wrong total.

Check the total, then adjust the largest stratum and state that you have.

Saying systematic sampling is biased because it is not random.

With a random start it is unbiased for most lists. Its specific risk is a periodic pattern matching the interval.

Offering "it is cheaper" as a full justification.

Tie the advantage to the scenario — no frame needed, population inaccessible, testing destroys the item.

Sampling

k = \tfrac{N}{n}

Systematic interval.

n_{i} = \tfrac{N_{i}}{N}\times n

Stratified sample size.

\text{census: every member}

Accurate but often impractical.

\text{sample: a subset}

Cheaper, but an estimate.

Stretch yourself

A company has 340 staff in production, 150 in sales and 110 in administration. A stratified sample of 60 is required. Find the sample size from each department, and explain why stratified sampling is preferable here to simple random sampling.

Hint — Find the sampling fraction first. Then think about what could go wrong if the smallest department were under-represented.

Reveal solution

SolutionTotal staff

= 340 + 150 + 110 = 600

, so the sampling fraction is

\tfrac{60}{600} = \tfrac{1}{10}

. Production: 34. Sales: 15. Administration: 11. These sum to 60 ✓. Stratified sampling is preferable because the three departments are likely to differ systematically in whatever is being measured — working conditions, pay, hours — and stratifying guarantees each is represented in proportion. A simple random sample of 60 could by chance include very few administration staff, whose 110 members are the smallest group, and the estimate for that group would then rest on a handful of people. Stratification removes that risk entirely rather than relying on it averaging out.

\blacksquare

Questions students ask

When is a census actually worth doing?

When the population is small, when complete accuracy is required, and when observing does not destroy the item. A school surveying its own 30 teachers can reasonably ask all of them.

Is systematic sampling random?

Partly. The starting point is chosen at random within the first interval, so it is not fully random — once the start is fixed, the whole sample is determined. It is treated as random in practice provided the list has no periodic structure.

What makes a sampling frame "usable"?

It must list every member of the population once, with no omissions or duplicates. An out-of-date register or a list that excludes part of the population introduces bias before sampling even begins.

Does a bigger sample fix bias?

No. A larger biased sample estimates the wrong thing more precisely. Only the selection method can remove bias; sample size only reduces random variation.

Key takeaways

A census covers everyone; a sample covers a subset drawn from a sampling frame.
Simple random sampling is unbiased but needs a full frame and effort.
Systematic sampling is quick but vulnerable to periodic patterns in the list.
Stratified sampling guarantees proportional representation of important subgroups.
Quota and opportunity sampling need no frame but are not random and carry specific biases.

How this fits the course

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