Is cluster sampling a form of random sampling?

Question

I learned in my stats class that cluster sampling is when you divide the population into subgroups called clusters, pick some of the clusters randomly, then select every member of the chosen clusters to be included in the sample. Is this a form of random sampling? For reference, my teacher also said that random sampling is when every member of the population has the same chance of being selected for the sample.

Answer 1

In fact, cluster sampling, as you have defined it, is a form of random sampling, and this can be shown as follows:

Suppose there are a total of R clusters, labeled C₁,C₂,...,C_R, and suppose that we intend to select exactly r of them randomly, then add each member of the selected clusters to our sample.

Take any individual member of the population, and let S denote the event that this member is ultimately selected for our sample. Of particular interest to us is P(S).

For each i = 1,2,...,R, define A_i to be the event that the member is in cluster C_i, and define B_i to be the event that C_i is one of the r clusters that is selected.

We have

P(S) = P(S and A₁) + P(S and A₂) + ... + P(S and A_R)

= P(A₁)•P(S|A₁) + P(A₂)•P(S|A₂) + ... + P(A_R)•P(S|A_R)

= P(A₁)•P(B₁) + P(A₂)•P(B₂) + ... + P(A_R)•P(B_R).

Now, each P(B_i) is equal to r/R, because the r clusters are selected randomly from the pool of R clusters. Factoring out r/R from each of the terms above gives

P(S) = r/R • [ P(A₁) + P(A₂) + ... + P(A_R) ].

The sum of the P(A_i) is 1 because the C_i partition the population, so that exactly one of the A_i must occur. This leaves just

P(S) = r/R.

Notice that this probability is constant. It does not depend on which member of the population was being considered. In other words, every member of the population has the same probability r/R of being selected for the sample. By definition, a sampling method which gives each member of the population the same probability of being selected for the sample is a random sample.

Answer 2

No this is not a form of random sampling because not every member of the total population has an equal chance of being randomly selected.

The members of the selected cluster have an equal chance of being selected at random, but that's only within that cluster not the total population.