Random Sampling

Simple random samples

In a simple random sample (SRS) of size $n$, every possible sample of $n$ units has the same chance of being the one selected. It is the basis for the other methods in this lesson.

To take one, give every unit in the population a number from 1 to $N$, then use a random number generator to pick $n$ different numbers. Drawing numbered slips of paper from a well-mixed container works too.

Almost every real sample is taken without replacement: once a unit is selected it is set aside, so it cannot be selected again, and a repeated random number is simply skipped. Sampling with replacement returns each unit to the population after it is selected, so the same unit can come up more than once. That fits a game show that puts each numbered ball back in the drum before the next draw, not a survey of people.

Stratified random samples

A stratified random sample first divides the whole population into non-overlapping groups called strata, based on a shared characteristic, so the units within a stratum are alike in that characteristic. Then a separate SRS is taken from each stratum, and the selected units are combined into one sample.

Stratifying pays off when the variable you care about probably differs from stratum to stratum. A survey about student parking should stratify by grade, because seniors drive and freshmen mostly do not. Every grade is guaranteed a place in the sample, and the estimate will not be thrown off by a random draw that happens to be heavy on freshmen.

Cluster random samples

A cluster random sample divides the population into smaller groups called clusters, takes an SRS of the clusters, and collects data from every unit in each selected cluster. Ideally each cluster is a small, mixed version of the population, so that the clusters are similar to one another.

Cluster sampling is usually picked because it is cheaper and easier to carry out; the clusters themselves are still chosen at random. To survey students about cafeteria food, a school could randomly select 6 of its 48 homerooms and survey everyone in them. Visiting 6 rooms is far easier than tracking down 150 students scattered across the building.

Systematic random samples

A systematic random sample picks a random starting point and then takes units at a fixed interval after it. To survey about 1 in 20 shoppers leaving a store, generate a random number from 1 to 20, say 7, and survey the 7th shopper, then the 27th, the 47th and so on.

Systematic samples are easy to run on a line of people or a list. They go wrong when the list repeats in a pattern that matches the interval: sampling every 7th day from a year of daily sales records lands on the same weekday every time.

Which method fits?

Each method has features that make it the better choice for some questions:

• An SRS works when you can list the whole population and reach anyone on the list.
• Stratify when the response probably differs between groups you can identify in advance, and you want every group represented.
• Use clusters when the population comes in natural groups and reaching scattered individuals would be slow or expensive.
• A systematic sample suits units that arrive in order, like customers at a door or items on an assembly line.

The justification always comes back to the context: stratify by grade, because seniors are much more likely than freshmen to drive.