Choosing by chance

In a simple random sample (SRS) of size $n$, every possible sample of $n$ units has the same chance of being the one selected. That is a stronger promise than "every unit has an equal chance", and it is the promise that the inference methods later in the course are built on.

The mechanics are deliberately boring:

1. Label. Number every unit in the population, 1 through $N$.
2. Generate. Use a random number generator (or slips of paper drawn from a container) to produce numbers in that range.
3. Select. Take the units whose numbers come up, skipping repeats until you have $n$ distinct units.

Skipping repeats makes this sampling without replacement: a selected unit is not returned, so it can be chosen at most once. Sampling with replacement returns each unit to the pool before the next draw, so the same unit can appear more than once. Surveys almost always sample without replacement; the distinction matters later when probability models enter.

A stratified random sample divides the whole population into non-overlapping groups called strata, built so the units within a stratum share an attribute: same grade level, same fare class, same region. Then a separate SRS is taken within each stratum, and the pieces are combined into one sample.

Stratifying earns its extra effort when the strata genuinely differ on the variable being studied. It buys two things:

1. Guaranteed representation. Every stratum contributes to the sample by construction. An SRS could, by chance, land lightly on a small group; a stratified design cannot.
2. Less sample-to-sample variability. When units within each stratum are alike on the response, each stratum's estimate is steady, and combining steady estimates gives a steadier total than one SRS of the same size.

Both benefits flow from the same condition: homogeneous within, different between. Stratifying by a variable unrelated to the question, say by the first letter of a last name, adds bookkeeping and buys nothing.

A cluster random sample also divides the population into groups, but uses them oppositely. Clusters work best when each one is a miniature of the whole population, heterogeneous inside and similar to one another. A random sample of clusters is selected, and then data are collected from every unit in each selected cluster. The randomness picks groups, not individuals, and the payoff is practical: surveying three whole buildings is cheaper than chasing an SRS scattered across a city.

The contrast with stratified sampling is exact, and exam questions live in it. Stratified: groups homogeneous inside, sample some units from every group. Cluster: groups heterogeneous inside, take every unit from some groups.

A systematic random sample uses a random starting point and then a fixed interval: start at unit 7, then take every 25th unit after it. The random start is what makes it a random sample. It is not an SRS, because most combinations of units can never occur together, but it spreads the sample evenly through the population and is easy to run on a production line or a long list.

Each random sampling method has characteristics that suit different situations, and "which design is this, and why was it a good idea" is a standing AP question. A quick fit guide:

1. SRS when the population is easy to list and reach, and no grouping matters.
2. Stratified when known groups differ on the response and each must be represented: opinions that split by grade level, usage that splits by age band.
3. Cluster when the population comes pre-packaged in mixed groups and travel or access is the cost: buildings, flights, crates.
4. Systematic when units flow past in sequence and a rhythm is convenient: every 25th phone off the line after a random start.

Two cautions close the topic. First, none of these designs is random without its chance mechanism: picking whoever is nearby, or letting people volunteer, is not random no matter how unpredictable it feels. Second, an SRS that happens to catch few members of some group is not thereby biased. Bias is a systematic property of the method; a fair mechanism producing an unlucky draw is variability, and the next topic takes up what bias really is.