How do you choose a sample that fairly represents a population, and what are the trade-offs of each sampling method?
Populations and samples, the census, sampling methods (simple random, systematic, stratified, quota and opportunity), their advantages and disadvantages, and the role of the large data set.
A focused answer to the OCR A-Level Mathematics A statistical sampling content, covering populations and samples, the census, simple random, systematic, stratified, quota and opportunity sampling, the advantages and disadvantages of each, sources of bias, and how the pre-release large data set is used.
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What this dot point is asking
OCR wants you to distinguish a population from a sample, know when a census is appropriate, describe and carry out the standard sampling methods (simple random, systematic, stratified, quota and opportunity), weigh their advantages and disadvantages, identify sources of bias, and understand how the pre-release large data set is used to ground the statistics content.
The answer
Population, census and sample
A population is the whole set of items of interest; a census observes every member of it. A sample is a subset used to make inferences about the population. A census is accurate but often too costly, slow or destructive (you cannot test every fuse to destruction), so sampling is usual.
The sampling methods
Random versus non-random
Simple random, systematic and stratified sampling all use randomness, so they tend to give representative, unbiased samples and support probability-based inference. Quota and opportunity sampling are non-random: they are quick and cheap but prone to bias, because the selection depends on who is convenient or on the interviewer's choices.
Examples in context
Choosing a method
The right method depends on what is known about the population and what resources are available. If a sampling frame (a list of the population) exists, simple random or systematic sampling is straightforward. If the population splits into meaningful groups of different sizes, stratified sampling keeps each group represented.
Sources of bias
Bias creeps in when some members are systematically more or less likely to be chosen. Common causes are an incomplete sampling frame (people omitted from the list), non-response (those who decline differ from those who answer), and self-selection (only the keen reply). Opportunity sampling at one place and time, as in the supermarket example, bakes in bias because it excludes everyone not there.
The large data set
OCR provides a single pre-release large data set for the life of the qualification. You explore it during the course so you know its variables, units, structure and any missing values. Statistics questions may quote extracts or summary statistics from it and reward familiarity with real-data judgement, such as spotting that a variable has gaps or outliers.
Try this
Q1. A factory makes items a day and wants a systematic sample of . State the sampling interval. [1 mark]
- Cue. .
Q2. Give one advantage of a census over a sample. [1 mark]
- Cue. It is completely accurate because every member of the population is counted (no sampling error).
Exam-style practice questions
Practice questions written in the style of OCR exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
OCR 20195 marksA college has students in three year groups: in Year 12, in Year 13 and part-time. A sample of is required. Explain how to take a stratified sample, and state one advantage and one disadvantage of stratified sampling.Show worked answer →
The sampling fraction is (M1). Apply it to each stratum: Year 12 gets , Year 13 gets , and part-time gets (A1).
Within each stratum, select that many students by simple random sampling, for example by numbering them and using random numbers (A1).
Advantage: the sample reflects the group structure of the population, so it is representative of each year group (B1). Disadvantage: you need to know the stratum of every member, and the method is more complex to organise than simple random sampling (B1).
Markers reward the sampling fraction, the correct stratum sizes summing to 60, random selection within strata, and a valid advantage and disadvantage.
OCR 20214 marksA researcher stands at a supermarket entrance one weekday morning and interviews the first shoppers who agree to take part. Identify the sampling method, and give two reasons why the sample may not represent all the supermarket's customers.Show worked answer →
This is opportunity (convenience) sampling, possibly with an element of self-selection because shoppers choose whether to take part (B1).
Reason 1: a weekday morning excludes people who shop at other times, such as those who work full-time, so the sample is biased towards a particular group (B1).
Reason 2: only those willing to stop are included, so the sample over-represents people with time and an interest in the topic (B1). A further point is that one entrance on one day is too narrow to generalise from (B1, any two reasons).
Markers reward naming opportunity sampling and two distinct, valid sources of bias explained in context.
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Sources & how we know this
- OCR A Level Mathematics A (H240) specification — OCR (2017)