Sampling methods, explained with examples
How you choose a sample decides whether your conclusions can be trusted. A good random sample lets you generalise to a whole population; a poor one can mislead, however large it is. Sampling questions appear on every AP Statistics exam.
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Population, sample and census
The population is the whole group you want to learn about, such as all students at a school. A sample is the part of the population you actually collect data from. A census collects data from every member of the population, which is often too slow, expensive or impossible. The goal of sampling is to choose a sample that represents the population well, so that conclusions from the sample apply to the population.
Simple random sample (SRS)
In a simple random sample of size n, every group of n individuals has the same chance of being chosen. To take an SRS: number every member of the population, then use a random number generator or table to choose n different numbers, ignoring repeats. A useful description on the exam: “Assign each of the 1200 students a unique number from 1 to 1200. Use a random number generator to select 100 different numbers. The students with those numbers form the sample.”
Stratified random sampling
Divide the population into groups, called strata, of similar individuals, then take a separate SRS from each stratum and combine them. For example, divide a school by year group and randomly select 25 students from each year. Stratifying works best when the strata are similar within themselves but different from each other on the variable of interest. It guarantees every group is represented and usually gives more precise estimates than an SRS of the same size.
Cluster sampling
Divide the population into groups called clusters, usually by location, then randomly select some whole clusters and survey everyone in them. For example, randomly choose 5 of a school's 40 form groups and survey every student in those 5. Cluster sampling is popular because it is cheap and convenient. It works best when each cluster is a mix that looks like the whole population.
A common exam question asks for the difference between stratified and cluster sampling. In stratified sampling, you take some individuals from every group. In cluster sampling, you take every individual from some groups.
Systematic random sampling
Choose a random starting point, then select every kth individual from an ordered list. To sample 100 from a list of 2000, k = 20: choose a random start between 1 and 20, such as 13, and then take 13, 33, 53, and so on. It is quick and spreads the sample across the list, but it can go wrong if the list has a repeating pattern that lines up with k.
Bad sampling methods
- Convenience sampling: choosing whoever is easiest to reach, such as friends or shoppers at one mall. It usually misses parts of the population.
- Voluntary response sampling: people choose themselves, for example by answering an online poll. Those with strong opinions, often negative, are far more likely to respond.
Both methods are biased: they tend to give answers that are systematically too high or too low, and a bigger sample does not fix the problem. A small random sample is often far more trustworthy than a huge self-selected one, a lesson from many famous polling failures.
Sources of bias
- Undercoverage: some groups are left out of the sampling process, such as people without home internet in an online survey.
- Nonresponse: people chosen for the sample cannot be contacted or refuse to take part, and they may differ from those who respond.
- Response bias: people give inaccurate answers, for example by lying about sensitive behaviour or misremembering.
- Wording of questions: leading or confusing questions push answers in one direction.
When describing bias on the AP exam, explain the likely direction: would the estimate be too high or too low, and why?
Worked examples
Example 1: naming the method
A researcher lists all 30 classrooms in a school, randomly picks 4 and surveys every student in them. Which method is this?
- Whole groups are selected and everyone in them is surveyed.
- Cluster sampling.
Example 2: choosing strata
A city wants to estimate average household income. Should it stratify by neighbourhood, and why?
- Incomes tend to be similar within a neighbourhood and different between neighbourhoods.
- Yes: stratifying by neighbourhood would ensure every area is represented and would give a more precise estimate than an SRS of the same size.
Example 3: describing bias
A radio host asks listeners to phone in with their view on a new tax. 85% of callers oppose it. Is 85% a good estimate for all residents?
- This is a voluntary response sample. People who feel strongly, especially those against the tax, are more likely to call.
- The true proportion opposed is probably lower than 85%, so the estimate is likely biased upwards.
Example 4: a systematic sample
Describe how to choose a systematic sample of 50 from an alphabetical list of 1000 club members.
- k = 1000/50 = 20. Randomly choose a starting number from 1 to 20, say 7.
- Select members 7, 27, 47, …, 987. That gives 50 members.
Example 5: undercoverage
A survey about public transport use is carried out by calling landline phones. Identify a source of bias and its likely effect.
- Many younger people have no landline, so they are left out: undercoverage.
- Younger people may use public transport more, so the survey would likely underestimate public transport use.
Common mistakes
- Confusing stratified sampling with cluster sampling.
- Saying a larger sample removes bias; it only reduces random variability.
- Naming a type of bias without explaining its likely direction in context.
- Describing an SRS without saying how the random numbers are used or that repeats are ignored.
Practice questions
Try each one before opening the answer.
Q1. Students are split by gender and 30 are randomly chosen from each. Which method?
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Q2. A website asks visitors to rate a product. What kind of sample is this?
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Q3. Why might stratified sampling be better than an SRS?
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Q4. Many people chosen for a survey do not reply. What is this bias called?
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Frequently asked questions
Is sampling on the AP Statistics exam?
Yes. Sampling methods and bias are in Unit 3, collecting data, and they appear in both multiple-choice and free-response questions.
Is sampling in A level Maths?
Yes. A level Maths statistics covers simple random, systematic, stratified, quota and opportunity sampling, with their advantages and disadvantages.
Key takeaways
- SRS: every group of size n is equally likely to be chosen.
- Stratified: SRS from every stratum; more precise when strata differ.
- Cluster: all individuals from randomly chosen clusters; cheap and convenient.
- Convenience and voluntary response samples are biased.
- Explain the direction of any bias, in context.