How can you perform a sample selected in such a way that each member of the population has an equal probability of being included?

Answer 1

The short answer is "random sample," but that, unfortunately, is neither specific nor complete. It is not specific because there are forms of random sampling where selection probabilities are not constant. It is not complete because there are many different ways to conduct random sampling with equal selection probabilities.

"Simple random sampling" occurs when you can perform a process that, for all practical purposes, behaves like writing down the identifier of each population member on a piece of paper, putting all the pieces into a box, mixing them thoroughly, and pulling out a few of them one by one (without replacing them in the box). Nowadays we use a computer to do this job, because it's faster and more reliable (it is notoriously difficult to mix pieces of paper perfectly randomly). The computer needs a complete list of all the population members: this is called a <i>sampling frame</i>.

Here is an example of random sampling that is not simple but still selects every population member with equal probability. Suppose you want to sample half the students in a classroom of 30. Ask them to line up. Flip a fair coin: if it's heads, pick the first, third, ..., 29th in line. If tails, pick the second, fourth, ..., 30th. Any individual student has a 50% chance of being part of the sample, so each student has an equal probability of being included. However, if you lined up the students boy-girl-boy-girl, etc., the samples themselves wouldn't look very random: they will either be mostly boys or mostly girls. It's still random though, because it's determined by the flip of a coin.

The example highlights a subtle but important property of a random sample: in many cases, you want the selection of population members to be <b>independent</b>. This means the probability of selecting one member is not affected by which other members are selected. In simple random sampling, independence holds; in the second example (a form of <i>gridded sampling</i>), there is complete dependence: no student can be chosen along with either of their neighbors in line, for instance.

Simple random sampling is ideal for many purposes but often cannot be carried out in practice because it is not feasible (you might not be able to construct a sampling frame) or costs too much. Often, more complicated procedures, such as <i>hierarchical sampling</i>, are carried out to overcome these limitations. (An example of hierarchical sampling is when an epidemiologist selects a city at random, then selects households at random within the city, then selects children at random within each household to study. Doing it this way can require much less travel than selecting children at random from all over the state.) These procedures might or might not select population members with equal probability. Usually the selection is not independent, either. When the probabilities are unequal, they can be figured out and used as <i>weights</i> in statistical analysis of the data. Results can also be adjusted for lack of independence.

A good, readable, non-technical introduction to sampling and simple random samples is the textbook <i>Statistics</i> by Freedman, Pisani, and Purves. Any edition is fine. Steven Thompson's book <i>Sampling</i> discusses dozens of different sampling procedures and explains the theory behind each one.