Representative Observation and Random sampling

Question

Suppose that we have a cross-sectional data set drawn by random sampling.

Then, the data may represent our population of interests.

We can consider two subpopulations: male group and female group.

Here, I want to draw a "representative" observation for each subpopulation.

If I choose a male "randomly" from the male subpopulation, can I say that this observation represents the subpopulation?

In a sentence, I want to draw a male who represent the subpopulation of men.

Then, is random drawing from the subpopulation an appropriate way?

Answer 1

Your question is not very clear. But: If the population (or subpopulation) you want a "representative" sample from is heterogeneous, then how can a sample of one unit be representative?

There are multiple problems:

• How do you define "representative"? In sampling theory this seems to be avoided, by discussing random sampling, not representativeness! So random sampling seems to be a substitute for "representative".

• With very small samples (especially $n=1$) there is a chance that the results of random sampling can be intuitively unreasonable, so in some cases a systematic selection might be better. But again, without a definition of what you mean with "representative", in your application, what to do?

See the classical book by D Cox: https://bookshop.org/books/planning-of-experiments/9780471574293 where this is discussed.