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This lesson covers sampling methods as required by the Edexcel A-Level Mathematics specification (9MA0), Paper 3 Section A -- Statistics. You need to understand the difference between a census and a sample, know the main types of sampling, and be able to discuss advantages and disadvantages of each method.
A census is a survey that collects data from every member of the population.
A sample is a survey that collects data from a subset of the population.
| Feature | Census | Sample |
|---|---|---|
| Coverage | Every member of the population | A selection of members |
| Accuracy | Gives a completely accurate result (no sampling error) | Subject to sampling error |
| Cost | Very expensive and time-consuming | Cheaper and quicker |
| Practicality | Often impractical for large populations | Practical for most situations |
| Destructive testing | Not possible (e.g. testing light bulbs to failure) | Suitable for destructive testing |
| Term | Definition |
|---|---|
| Population | The whole set of items that are of interest |
| Sample | A subset of the population used to collect data |
| Sampling unit | Each individual member of the population that can be sampled |
| Sampling frame | A list of all sampling units (e.g. the electoral register, school roll) |
| Statistic | A quantity calculated from a sample (e.g. sample mean) |
| Parameter | A quantity that describes a characteristic of the whole population (e.g. population mean) |
Exam Tip: A census gives accurate results but is time-consuming, expensive, and sometimes destructive (e.g. testing the lifetime of batteries would destroy every single one). A sample is quicker, cheaper, and the only option when testing is destructive.
A sampling frame is a list (or database) of every member of the population from which the sample is drawn. For example:
For a sample to be representative, the sampling frame must be accurate and up to date. If some members of the population are missing from the frame, the sample will be biased.
In simple random sampling, every member of the population has an equal chance of being selected. Every possible sample of a given size is equally likely.
In systematic sampling, you select members at regular intervals from an ordered list.
A factory produces 500 items per day. You want a sample of 25 items.
In stratified sampling, you divide the population into strata (distinct subgroups that do not overlap), then take a random sample from each stratum in proportion to the stratum's size.
Number from stratum = (number in stratum / total population) x total sample size
A school has 600 students: 200 in Year 12 and 400 in Year 13. You want a stratified sample of 60 students.
Then use simple random sampling within each year group to select the required number.
In quota sampling, the interviewer is given a quota -- a specified number of people to survey from each subgroup. The interviewer then selects people who fit the required profile until the quota is filled.
A market researcher needs to survey 100 people: 50 males and 50 females. The researcher stands in a shopping centre and asks people until they have 50 of each.
In opportunity sampling (also called convenience sampling), you simply select people who are available at the time of the study. You use whoever happens to be around.
A student stands outside the school canteen at lunchtime and asks the first 30 people who walk past to complete a questionnaire.
| Method | Random? | Needs sampling frame? | Proportional? | Main advantage | Main disadvantage |
|---|---|---|---|---|---|
| Simple random | Yes | Yes | Not guaranteed | Bias-free | Needs complete list |
| Systematic | Partially (random start) | Yes | Not guaranteed | Even spread | Patterns can cause bias |
| Stratified | Yes (within strata) | Yes (for each stratum) | Yes | Proportional representation | Complex to set up |
| Quota | No | No | Yes (by design) | No sampling frame needed | Selection bias |
| Opportunity | No | No | No | Quick and easy | Very biased |
When designing a sampling method, you should consider the type of data being collected:
Exam Tip: The exam may ask you to recommend a sampling method for a given scenario. Consider whether a sampling frame is available, whether the population can be divided into clear strata, whether the situation requires random selection, and any practical constraints (cost, time, access).
Type 1: "Explain why a census might not be appropriate."
Answer: A census surveys every member of the population. This is expensive, time-consuming, and impractical for large populations. If the testing process is destructive, a census would destroy the entire population.
Type 2: "Describe how to take a stratified sample of 50 from a population of 1000 with 600 in group A and 400 in group B."
Answer: Divide the population into the two strata. Group A: (600/1000) x 50 = 30. Group B: (400/1000) x 50 = 20. Use simple random sampling within each group to select 30 from A and 20 from B.
Type 3: "Give one advantage and one disadvantage of opportunity sampling."
Answer: Advantage -- it is quick and easy to carry out. Disadvantage -- it is likely to be biased and not representative of the population.
Edexcel 9MA0-03 specification, Paper 3 -- Statistics and Mechanics, Section 1: Statistical sampling. The specification requires students to "understand and use sampling techniques, including simple random sampling and opportunity sampling," and to "select or critique sampling techniques in the context of solving a statistical problem, including understanding that different samples can lead to different conclusions about the population." Sampling underwrites every later statistics section: Section 2 (data presentation and interpretation), Section 3 (probability), Section 4 (statistical distributions) and Section 5 (statistical hypothesis testing) all assume a sample drawn from a population. Sampling theory is also examined synoptically through the Edexcel Large Data Set (LDS) -- a published weather data set from UK and overseas Met Office stations that candidates are expected to have engaged with prior to the exam.
Question (8 marks):
A school of N=1,200 students is divided into year groups of sizes 320 (Y7), 300 (Y8), 260 (Y9), 180 (Y10) and 140 (Y11). The headteacher wants a sample of 60 students to investigate satisfaction with school meals.
(a) State, with a reason, an appropriate sampling method. (2)
(b) Using your chosen method, calculate how many students should be sampled from each year group, and describe in detail how the sample should be selected. (4)
(c) Identify one source of bias that may still affect the results, even when the chosen sampling method is applied correctly. (2)
Solution with mark scheme:
(a) Stratified sampling is appropriate because the population divides naturally into non-overlapping strata (year groups) which are likely to differ in their views -- older students may have different food preferences from younger ones -- and the strata sizes are known.
B1 -- naming stratified sampling. B1 -- a reason that engages with stratification (heterogeneity between strata, homogeneity within). A reason such as "it is fair" earns nothing; the reason must reference the structure of the population.
(b) Step 1 -- compute the sampling fraction.
f=120060=201=0.05
M1 -- correct sampling fraction (or equivalent proportional reasoning).
Step 2 -- apply to each stratum.
| Year | Size | Allocation | Rounded |
|---|---|---|---|
| Y7 | 320 | 320×0.05=16 | 16 |
| Y8 | 300 | 300×0.05=15 | 15 |
| Y9 | 260 | 260×0.05=13 | 13 |
| Y10 | 180 | 180×0.05=9 | 9 |
| Y11 | 140 | 140×0.05=7 | 7 |
| Total | 1200 | 60 |
A1 -- all five allocations correct and summing to 60.
Step 3 -- describe selection within each stratum. Within each year group, assign every student a unique ID number from 1 to the stratum size. Use a random-number generator (or random-number tables) to draw the required number of distinct IDs without replacement; the selected IDs identify the students in the sample.
M1 -- explicit random selection within each stratum (the word "random" is essential). A1 -- adequate detail: numbered list, random-number generator, no replacement.
(c) Non-response bias. Even if the 60 students are correctly selected, those who choose not to return the questionnaire -- or who skip the dining hall on the survey day -- are likely to differ systematically from those who respond (perhaps the most dissatisfied avoid school meals altogether). The achieved sample is no longer representative.
B1 -- naming a plausible source of residual bias (non-response, response/social-desirability, sampling-frame omissions, volunteer effect). B1 -- explaining why this distorts representativeness.
Total: 8 marks.
Question (6 marks): A market-research company wants to estimate the mean weekly grocery spend of households in a town of approximately 8{,}000 households. The company has access to the council electoral register, listing every adult resident.
(a) Describe how a systematic sample of 200 households could be drawn from this register, making clear any assumptions you rely on. (3)
(b) State one advantage and one disadvantage of systematic sampling compared with simple random sampling in this context. (2)
(c) Suggest one reason why the resulting sample may not be representative of the town's households, even if the sampling procedure is followed correctly. (1)
Mark scheme decomposition by AO:
(a)
(b)
(c) B1 (AO3.5a) -- sampling-frame coverage: the electoral register omits households where no adult is registered (transient renters, students, recent arrivals, those opting out), so the achieved sample under-represents these groups.
Total: 6 marks split AO3 = 6. This is an entirely AO3-dominated question -- modelling and critique of a real-world sampling procedure is the AO3 territory of Paper 3 statistics.
Connects to:
Section 2 -- Data presentation and interpretation: the histograms, box plots and summary statistics of Section 2 are computed from samples. A poorly drawn sample produces well-presented but misleading visualisations -- the integrity of the chart depends on the integrity of the sample.
The Edexcel Large Data Set (LDS): the LDS is itself a sample -- daily weather observations from a fixed set of UK and overseas Met Office stations over a defined period. Candidates are expected to know that the LDS is not the population of all UK weather, and to recognise where its sampling structure (fixed stations, fixed time window) limits what can be inferred. Exam questions routinely ask candidates to critique conclusions drawn from the LDS by reference to its sampling design.
Section 5 -- Hypothesis testing: the binomial test H0:p=p0 versus H1:p=p0 assumes the sample is drawn from a Bernoulli process -- i.e. that observations are independent and identically distributed. This assumption fails for cluster or quota samples without correction; the inferential machinery breaks if the sample is unrepresentative.
Section 4 -- Statistical distributions: the normal-distribution model in 9MA0 assumes random sampling. The Central Limit Theorem (mentioned implicitly via large-sample normal approximations) requires independent draws -- exactly what simple random sampling provides and what opportunity or quota sampling does not.
Bias minimisation across the syllabus: the language of "representative sample" recurs in Year 13 statistical inference (correlation, regression, the product-moment correlation coefficient r). A correlation computed on a biased sample is uninterpretable as a population correlation -- Section 1 sampling is the foundation for Section 5 inference.
Sampling questions on 9MA0 Paper 3 split AO marks unusually heavily toward AO3:
| AO | Typical share | Earned by |
|---|---|---|
| AO1 (knowledge / procedure) | 20-30% | Naming sampling methods correctly; computing stratum allocations; stating sampling-interval formulae |
| AO2 (reasoning / interpretation) | 20-30% | Justifying why a chosen method suits the population; interpreting the consequences of a stated bias |
| AO3 (problem-solving / modelling) | 45-60% | Critiquing a sampling procedure in context; identifying residual bias; recommending and describing improvements |
Examiner-rewarded phrasing: "the population divides into non-overlapping strata which are believed to differ"; "every member of the population has a known, non-zero probability of selection"; "a random starting point is chosen between 1 and k inclusive"; "the sampling frame omits ..., so the sample is not representative of the population". Phrases that lose marks: "it is fair" (vacuous); "the sample is random because the people were chosen randomly" (circular); "this method has no bias" (unsupportable as written).
A specific Edexcel pattern to watch: questions phrased "describe how" require a procedure -- numbered or sequenced steps -- not a definition. Candidates who write the textbook definition of stratified sampling instead of describing how to apply it to the given population routinely score B1 out of 4.
Question: A researcher stops the first 30 shoppers leaving a supermarket and asks each their age. Name this sampling method and give one reason it may produce a biased estimate of the mean age of supermarket customers.
Grade C response (~180 words):
This is opportunity sampling because the researcher uses whoever is conveniently available. It may be biased because the first 30 people leaving may not be typical -- they might all be there at the same time of day, like working adults during a lunch break, so older retired people or younger school children would be missed. So the mean age of the sample would not match the mean age of all customers across the day.
Examiner commentary: Full marks (3/3). The method is correctly named (B1), and the reason engages with why the convenience aspect produces unrepresentativeness rather than just asserting "it is biased". The candidate gives a concrete mechanism -- time-of-day clustering -- which is the AO3 reasoning examiners want. The answer is concise and on-task. A weaker answer would write "opportunity sampling is biased because it is not random," which restates the property without explaining the cause.
Grade A response (~240 words):*
The method is opportunity sampling (also called convenience sampling): the researcher selects whoever is available at a single time and place, with no underlying random mechanism.
The sample is likely to be biased for several compounding reasons. First, time bias: the 30 shoppers leaving at one moment are not representative of the full daily flow. Pensioners often shop mid-morning; commuters after 6 p.m.; parents with young children mid-afternoon. The chosen window over-represents one demographic. Second, location bias: customers leaving the front entrance differ systematically from those using a side or car-park exit. Third, selection mechanics: the researcher's own subconscious choices -- approaching shoppers who appear willing -- exclude those in a hurry or on the phone, who may differ in age. The sample mean age will therefore be a biased estimator of the population mean age of supermarket customers; the bias direction depends on the time slot but is not zero in expectation.
Examiner commentary: Full marks (3/3) plus signal of A* depth. The candidate names three independent bias mechanisms (time, location, mechanics) rather than one, frames the conclusion in inferential language ("biased estimator of the population mean"), and uses the technical phrase "in expectation". This is the kind of answer that, on a longer 6- or 8-mark question, would secure the AO3 marks reserved for sophisticated critique.
Question: A factory produces 5{,}000 components per shift in batches of 50. Quality control wants a sample of 100 components per shift to estimate the proportion of defective items. Compare and contrast simple random sampling, systematic sampling, and cluster sampling for this purpose, recommending one method with justification.
Grade B response (~250 words):
Simple random sampling: assign each component an ID from 1 to 5000, use a random-number generator to pick 100 without replacement. This is fair but in a factory setting it is impractical because components are produced sequentially and cannot easily be retrieved from arbitrary positions.
Systematic sampling: every 50th component (since 5000/100=50), starting from a randomly chosen position 1-50. This is easy to implement on a production line. The risk is that if defects occur on a periodic cycle matching the sampling interval (every 50th component), the systematic sample misses them entirely.
Cluster sampling: pick whole batches of 50 at random and inspect everything in them. Two batches give 100 components. This is fast but if defects cluster within a batch (a faulty machine producing one bad batch), the variance of the estimate is much higher than under random or systematic sampling.
I recommend systematic sampling. It is the easiest to implement on a production line and the periodicity risk is small if defect causes are independent of position.
Examiner commentary: 5/6. All three methods are correctly described and one is recommended with a reasoned justification. The candidate loses one mark because the comparison of variance properties is informal -- an A* answer would note that cluster sampling typically inflates the variance of the estimator relative to simple random sampling because within-cluster correlation reduces effective sample size.
Grade A response (~290 words):*
Simple random sampling (SRS). Each of the 5{,}000 components has equal probability 5000100=0.02 of selection. SRS is the gold standard for unbiasedness and gives the standard variance for the proportion estimator. However, retrieving 100 individually-selected components from a continuous production line is operationally costly.
Systematic sampling. With sampling interval k=50 and a uniformly random start r∈{1,…,50}, take components at positions r,r+50,r+100,…. Operationally trivial -- one component per batch automatically. The danger is periodic bias: if any defect-generating mechanism has period 50 (e.g. a 50-cycle tool wear pattern), the sample either always hits or always misses defects, producing a biased estimator. Mitigation: shuffle the start each shift.
Cluster sampling. Randomly pick 2 of the 100 batches and inspect all 50 components in each. Fast in the field (the inspector handles two intact batches), but the design effect is unfavourable: if defects cluster within a batch (a single rogue machine), the within-cluster correlation ρ>0 inflates the variance of the proportion estimator by a factor of approximately 1+(m−1)ρ where m=50 is the cluster size.
Recommendation: systematic sampling. It balances representativeness (every batch contributes) against operational simplicity. Periodic-bias risk is manageable provided the sampling interval is not synchronised with any known production cycle. SRS is theoretically purer but logistically expensive; cluster sampling is the cheapest but inferentially weakest.
Examiner commentary: Full marks (6/6). The candidate uses the technical vocabulary of survey sampling -- design effect, within-cluster correlation, variance inflation -- and gives a balanced recommendation rather than a one-line preference. The mathematical detail (1+(m−1)ρ) signals undergraduate-level confidence.
Question: A government health department wants to estimate the mean daily fruit-and-vegetable consumption of UK adults. They propose to select 1{,}000 respondents through an online panel, stratified by region.
(a) Describe how the stratified online-panel sample should be drawn, including how stratum sizes should be allocated. (3)
(b) Identify and explain three distinct sources of bias that may affect this sample even if the stratification is implemented correctly. (4)
(c) Suggest one improvement to the design that would reduce bias, and justify your suggestion. (2)
Grade A response (~370 words):*
(a) Use national population estimates to find the adult population Ni in each of the k standard regions (e.g. North East, North West, ..., South West, London, Wales, Scotland, Northern Ireland). Compute the sampling fraction f=1000/N where N=∑Ni. Allocate ni=f⋅Ni respondents per region, rounding to integers and adjusting the largest stratum to enforce ∑ni=1000. Within each region, randomly select ni respondents from the panel using a random-number generator on panel IDs. (M1 sampling fraction; M1 proportional allocation; A1 within-stratum random selection.)
(b) Coverage bias. Online panels recruit people with internet access and a willingness to register on a survey site. Adults without broadband (older, lower-income, rural) are systematically under-represented. The achieved sample is the population of online-panel members, not the population of UK adults.
Self-selection / volunteer bias. Panel members opt in. Those who join may differ systematically from non-joiners (more time, more interest in surveys, possibly stronger views on health) -- they may already eat more fruit and vegetables on average than the general population.
Reporting / social-desirability bias. Self-reported dietary intake is consistently inflated in surveys: respondents over-report "good" foods and under-report "bad" foods to present favourably, even anonymously. (B1 each, max 3 from the list -- coverage / volunteer / response / non-response / panel-conditioning.)
B1 for explaining why each bias distorts the population mean.
(c) Improvement: combine the online panel with face-to-face interviews drawn from a household-address sample for offline households. This addresses the coverage bias directly by including the offline population, and the dual-mode design lets researchers calibrate online responses against a probability-based reference sample. The cost is operational complexity, but the inferential gain is substantial. (B1 named improvement; B1 justified by reference to a specific bias.)
Examiner commentary: Full marks (9/9). The answer demonstrates exam-craft (the three biases are distinct and named), AO3 modelling (an improvement is proposed and justified) and synoptic awareness (calibration to a probability sample is the language of professional survey methodology). This is the level of writing that wins the top A1/B1 marks consistently across Section 1.
The errors that distinguish A from A* on sampling questions:
Confusing stratified with systematic sampling. Stratified divides the population into groups first, then samples within each. Systematic samples every kth element from an ordered list. Candidates often write "stratified" when they mean "systematic" because both involve a mechanical procedure -- but the underlying logic is opposite (within-group homogeneity vs ordered-list traversal).
Treating sample size as the cause of bias. Bias is a property of the method, not the size. A biased method with n=10,000 remains biased; a small SRS sample with n=30 is unbiased (just imprecise). A* candidates separate bias (systematic error) from variance (random error) and recognise that throwing more data at a biased procedure does not help.
Forgetting that random selection requires a sampling frame. "Pick 50 people at random" is meaningless without a list to draw from. The sampling frame is the operational definition of the population -- and any mismatch between frame and target population is a coverage problem the random mechanism cannot fix.
Believing non-response is fixable by replacement. Replacing a non-respondent with the next person on the list does not eliminate non-response bias -- it disguises it. The non-respondents are systematically different; replacing them silently introduces the same bias plus over-representation of the easy-to-reach.
Claiming cluster sampling is "more efficient" without qualification. Cluster sampling is operationally efficient (fewer locations to visit) but statistically inefficient (within-cluster correlation inflates variance). The correct phrasing is "more cost-efficient per unit but less statistically efficient per unit".
Quota sampling as "stratified-without-randomness". Quota sampling fixes the target counts per group (like stratified) but lets interviewers fill quotas any way they choose (unlike stratified, which requires random selection within strata). Quota sampling is non-probabilistic and inferentially weaker; treating it as equivalent to stratified is a common A-grade error.
Equating "large data set" with "population". The Edexcel LDS is a designed sample of weather stations and time periods -- not the population of UK weather. Conclusions drawn from the LDS apply only to the sampled stations and seasons; extrapolating to "all UK weather" is exactly the inferential overreach Edexcel rewards critique of.
Three patterns repeatedly cost candidates marks on Paper 3 sampling questions. They are about precision of language, not technique.
This pattern is endemic to Section 1 questions: candidates know the methods, lose marks on procedural specificity.
Sampling theory at A-Level points directly toward several undergraduate trajectories:
Oxbridge interview prompt: "A pollster gets 60% support for candidate A from a random sample of 1{,}000 voters. Three days later they get 58% from a different random sample of 1{,}000. Has support fallen? How would you decide?"
A common A* question on Paper 3 gives unbalanced strata and asks for a stratified sample with proportional allocation, sometimes embedding a rounding subtlety.
Worked example: A college has 1{,}840 students distributed across four faculties: Arts (420), Sciences (610), Humanities (510), Engineering (300). A stratified sample of size 100 is required.
Compute the sampling fraction f=100/1840=5/92≈0.05435.
| Faculty | Size Ni | Ni⋅f | Rounded |
|---|---|---|---|
| Arts | 420 | 22.83 | 23 |
| Sciences | 610 | 33.15 | 33 |
| Humanities | 510 | 27.72 | 28 |
| Engineering | 300 | 16.30 | 16 |
| Total | 1840 | 100.00 | 100 |
The rounded allocations sum to 23+33+28+16=100 exactly -- but this is luck, not guarantee. When rounding produces ∑ni=99 or 101, the convention is to adjust the largest stratum by ±1 to enforce the required total; this minimises the proportional distortion.
Within each faculty, every student is given an ID; random-number generation produces ni distinct IDs without replacement; the chosen students form the sample.
Why A candidates spot the subtlety:* they always check that the rounded allocations sum to the required total, and they know to adjust the largest stratum first when they don't. They also write the sampling fraction as an exact rational 5/92 alongside the decimal, signalling a respect for exactness that examiners reward when allocations are awkward.
A subtlety: proportional allocation is not the only valid stratified design. Optimal (Neyman) allocation weights strata by both size and within-stratum standard deviation σi, allocating more to noisier strata. A-Level Edexcel uses proportional allocation exclusively, but mentioning Neyman allocation in an Oxbridge interview signals reading beyond the syllabus.
This content is aligned with the Pearson Edexcel GCE A Level Mathematics (9MA0) specification, Paper 3 -- Statistics and Mechanics, Section 1: Statistical sampling. For the most accurate and up-to-date information, please refer to the official Pearson Edexcel specification document.
graph TD
A["Population<br/>and research question"] --> B{"Sampling frame<br/>available?"}
B -->|"Yes, ordered"| C["Systematic sampling<br/>interval k, random start"]
B -->|"Yes, with strata"| D["Stratified sampling<br/>proportional allocation"]
B -->|"No frame"| E["Opportunity / quota<br/>(non-probabilistic)"]
C --> F["Check for<br/>bias and<br/>representativeness"]
D --> F
E --> F
style D fill:#27ae60,color:#fff
style F fill:#3498db,color:#fff