How to choose a sampling method

Start with what you know and what it costs

Every sampling decision comes down to two facts about your situation. The first is what you already know about the population: is there a trait that splits it into groups, and does that trait move the answer you are after? The second is what it costs to reach people: do you have a list of individuals, and are they scattered or already gathered into places you can visit?

Those two facts pick the method almost by themselves. Knowing a trait that matters pushes you toward stratifying. Having no list but plenty of intact groups pushes you toward clusters. A clean list and a reachable population leave you with the simplest option, a simple random sample.

The AP course splits this into two jobs, and both appear on the exam. Topic 1.11 asks you to identify a sampling method from a description of a study, and then to justify why a method is appropriate for the question being asked. The course states that each random sampling method has different characteristics that make it more appropriate depending on the population and the question, so there is no single best method to memorize.

What each method does to the population

There are five options, and what separates them is what they do to the population before anyone gets picked.

• Census. You record information from all individuals in the population. Nothing is sampled, so there is no sampling variability to account for.
• Simple random sample (SRS). No split at all. You draw $n$ individuals, where $n$ is the sample size, from one list, so that every possible sample of size $n$ has the same chance of being selected.
• Stratified random sample. You divide all individuals into non-overlapping groups called strata, based on one or more shared characteristics, then take an SRS inside each stratum and combine those pieces into one sample.
• Cluster random sample. You divide the population into groups called clusters, take an SRS of the clusters, and collect data from every individual in the selected clusters.
• Systematic random sample. You put the population in order, choose a random starting point, and take every $k$-th unit from there, where $k$ is a fixed interval you set in advance.

Only the census skips randomness on purpose. The other four all use a random mechanism, usually a random number generator, and that mechanism is what keeps the sample from being systematically off. A method with no random mechanism anywhere in it is not on this list for a reason.

The decision table

Read down the left column until a row matches your situation, then take the method beside it.

Two rows can both look true, and that is normal. When stratifying and clustering both seem workable, ask which constraint is actually binding: if the problem is precision, stratify; if the problem is cost or a missing list, cluster.

Stratified vs cluster, stated plainly

This is the pair that decides the most exam questions, and one sentence separates them.

Strata are internally similar, and you sample within all of them. Clusters are internally diverse, and you take whole ones.

Everything else follows from that. Strata are built from a shared attribute, so the individuals inside one stratum resemble each other and differ from those in other strata. Because you take an SRS inside every stratum, no stratum can be missed, and the differences between strata stop contributing to the sampling error.

Clusters are built the other way round. A good cluster mirrors the variety in the whole population, and the clusters resemble one another, so any few of them stand in for the rest. You choose clusters at random and then measure everyone inside the chosen ones, which is why a cluster sample can be run without ever building a list of individuals.

The tell in a scenario is what happens after the split. If the study samples a fraction of every group, it is stratified. If it picks a few whole groups and measures everyone in them, it is cluster. What is a cluster sample works through the second case, and simple random vs stratified sampling sets all four random methods side by side.

When a census beats any sample

A census records information from all individuals in the population, so it answers the question exactly instead of estimating it. That is the right call when the population is small, already listed, and cheap to contact: the 38 teachers in one building, or the 12 machines on one production line.

It stops being the right call quickly. A census of a large population costs more than a sample, takes longer, and often produces worse data, because rushed collection introduces measurement errors a careful sample would avoid. Nonresponse does not vanish either. An attempted census that only 40% of people answer is not a census; it is a sample of the people willing to answer.

The tradeoff between counting everyone and estimating from a few is laid out in census vs sample survey.

Two methods that are not really choices

Convenience sampling and voluntary response sampling turn up in scenarios constantly, and neither is a method you would pick on purpose. Both are nonrandom: they do not use random chance to select individuals, so they carry potential bias by construction.

Convenience sampling takes whoever is easy to reach. A reporter interviewing shoppers outside one store at 10am on a Tuesday gets people who shop there, at that hour, on a weekday. The people who are hardest to reach usually differ from the people who are easiest, and sampling more of the easy ones does not repair that.

Voluntary response sampling takes whoever volunteers. A website poll or a mail-in comment card produces a sample made up entirely of volunteers, and people with strong opinions volunteer at higher rates than people who are satisfied or indifferent. That is voluntary response bias, and it usually pushes an estimate of dissatisfaction upward.

Bias in a sampling method is a systematic error in the procedure that makes a statistic come out consistently larger or consistently smaller than the parameter it estimates. That is exactly why a bigger sample does not help: a larger convenience sample measures the wrong thing more precisely, an argument worked through in does a bigger sample fix bias. The other named problems in topic 1.12, undercoverage, nonresponse, and response bias, are sorted out in how to identify the type of bias.

Write the justification, not just the name

Naming the method is one skill; justifying it is a separate one, and questions ask for both. A justification that scores has three short parts.

1. Name the method in the language the course uses: simple random sample, stratified random sample, cluster random sample, systematic random sample, or census.
2. Point at the feature of the situation that made it right, such as a trait that varies across known groups or a list of individuals that does not exist.
3. Say what it buys for this question: guaranteed representation of each group, lower cost, or less variability in the estimate.

A finished sentence reads like this. Stratified random sampling is appropriate here because textbook spending differs sharply from one college to another, and sampling within every college guarantees all four colleges appear while reducing the variability of the estimated mean.

One thing a justification can never claim is that the sampling method makes the results causal. Sampling decides who is in the study, and therefore who you may generalize to; it says nothing about cause. That job belongs to random assignment, which is the subject of can you generalize these results and experiments vs observational studies.

To drill the naming skill against fresh scenarios, work through sampling methods practice. The official course framework lives at AP Central.