AP Stats 1.11: Random Sampling

Replacement and the simple random sample

Topic 1.11 is about how you choose the individuals in a sample, starting with replacement. Sampling without replacement means a unit can be chosen only once, because it is not returned to the population before the next draw. Sampling with replacement returns each unit, so it can be chosen more than once. Most real surveys sample without replacement.

The building block for the rest is the simple random sample (SRS). In an SRS of size $n$, every possible sample of size $n$ has the same chance of being selected. You can carry one out with a random number generator or by drawing numbered slips of paper, and it underlies the other sampling methods. The three named methods that follow are all ways of building randomness into who gets chosen.

Stratified, cluster, and systematic samples

Three more named methods build on random selection.

• A stratified random sample splits the population into non-overlapping groups, called strata, of individuals that share an attribute (homogeneous groups), then takes an SRS within each stratum and combines them. Splitting by grade level is a common example.
• A cluster random sample splits the population into clusters that each ideally mirror the whole population, takes an SRS of whole clusters, and collects data from every individual in the chosen clusters.
• A systematic random sample picks a random starting point, then selects every $k$th individual at a fixed interval.

See simple random vs stratified sampling for the two that are most often confused.

Choosing a method

No single method is best for every study. Each random sampling method has characteristics that make it more appropriate depending on the population and the question being investigated. Stratifying helps when groups differ and you want each represented; clustering saves effort when the population is spread across natural groups; a systematic sample is simple when individuals arrive in a line or list.

What they share is the use of chance to select individuals, which is what makes it valid to generalize the result to the population. On the exam, name the method and give a short reason tied to the population, rather than only stating what was done. Random selection controls who is in the sample; it is a separate idea from the random assignment of treatments in an experiment, covered in topic 1.13.