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I came across this test question from an introductory statistics course for undergraduates in biology. The solutions are in square brackets.

Which cases are possible?

  1. The sample is larger than the population. [False]
  2. The sample equals the population. [True]
  3. The sample is smaller than the population. [True]
  4. The sample is an empty set. [False]

I can't wrap my head around the solutions.

If we define the sample as a subset of the population, then 1. is false, and 2. and 3. are true indeed. But also 4. should be true because the empty set is a subset of any set.

However, if we define the sample as a proper subset of the population, then 1. and 2. are false, and 3. and 4. are true, assuming that the population is nonempty.

Quibbling over the empty set might be too pedantic. The main focus of my question is case 2., and my intuition suggests that it should be false. There is no sampling involved in the literal sense if one can examine the whole population, isn't there?

Additionally, I suspect that biologists may tend to inadvertently associate the word population with the concept of biological populations. And in principle, it's possible to examine every individual of a biological population. I'm also a biologist, but instead of biologists, statisticians have taught me statistics. And my recollection is that the concept of statistical populations is much more abstract. I'm not even sure whether it is meaningful to say something like examining every element of a statistical population.
I remember a remark from one of my teachers. In response to a nontrivial question (which has escaped my mind), they said something along the lines of "Well, we usually don't confess this at introductory courses but let me tell you: the statistical population doesn't really exist." Unfortunately, their explanation was over my head, so I can't recall it.

So does it make sense to say that the sample can equal the population, or it does not? And if not, then how to conceive statistical populations? References to relevant literature are much appreciated.

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  • $\begingroup$ If you can sample 10 of the 12 marbles in an urn, what is stopping you from sampling all 12? $\endgroup$
    – BruceET
    Feb 6, 2022 at 3:19
  • $\begingroup$ The given answer to (1) is generally wrong and, as you note, the given answer to (4) is mathematically incorrect (although that's a nitpick). It comes down to what this particular book's definition of "sample" is. Many samples are taken with replacement and, in such cases, they can be much larger than the population. Sound strange? It happens all the time when people bootstrap small datasets. BTW, often there are problems with defining a "population," but in some cases it is well-defined and obviously exists. Once again, bootstrapping supplies an example. $\endgroup$
    – whuber ♦
    Feb 6, 2022 at 14:21

1 Answer 1

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It can realistically be the case that you end up with the whole population. There's a question then as to whether you call it a sample, but I think it's reasonable to call it a sample (at least) when you didn't know in advance you would get the whole population.

Suppose you have a plan for spending a week sampling some area. You don't know in advance how much you'll get done in that time -- it might depend on weather or if you're sampling things that move around, it might depend on where they move. You could find that you get to all the streams or all the tall trees or the whole known population of takahē in the area. Your plan didn't call for (necessarily) getting the whole population, but your sample ended with the whole population in it. In that setting I think it would be very natural to still call what you have a sample.

Given that possibility, you could imagine there might be other reasons to call the data you have a sample even if it is the whole population. So answer 2 is not wildly unreasonable. It is, however, just a choice about the precise meaning of the word "sample". I would also argue that it's reasonable to define 'sample' either so that 4 is true, or so that 4 is untrue. I would not consider it reasonable to define 'sample' so that 1 is true. In a test, the correct answers would depend on precisely how the course had defined 'sample', and that's why I don't really like this sort of question.

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    $\begingroup$ A sample of an entire population is a census. $\endgroup$
    – Alexis
    Feb 6, 2022 at 2:12
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    $\begingroup$ A sample that is intended to collect the entire population can indeed be called a census. I've never seen a proper probability sampling procedure that has some small probability of collecting the whole population be called a census. $\endgroup$
    – Thomas Lumley
    Feb 6, 2022 at 2:15
  • $\begingroup$ Yeah, I agree with that. I have also seen enumeration for a sample containing whole population. $\endgroup$
    – Alexis
    Feb 6, 2022 at 2:17
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    $\begingroup$ Wouldn't it be problematic to define something as objective as a sample or census in terms of intentions? A process that yields a dataset either creates a census or not, regardless of what an investigator might have been thinking or wishing. As far as (1) goes, we should heed your initial remark that it depends on just what the book defines to be a "sample." If its definition includes sampling with replacement, then certainly the sample can be more numerous than the population. $\endgroup$
    – whuber ♦
    Feb 6, 2022 at 14:24
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    $\begingroup$ I have never seen a frequentist text that even attempted to model intentions, much less incorporated them in an analysis! I suppose you are equating "intentions" with "sample design." But any sample design that results in obtaining a census would ordinarily be considered a census. $\endgroup$
    – whuber ♦
    Feb 7, 2022 at 14:50

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