# Another question on population vs. sample

The problem that bugs me can be boiled down to: "What is population? (Really)." This is my thought experiment I've dealing with for a few days.

There is a researcher A, who gathers data from $$N = 20$$ observations on some variable. He gets $$\bar{x_A} = 17.21$$. He thinks that he can't measure all the population, 'cause "the amount of money, time" is too much to do that (the usual story, lecturers tell the students).

But there is also a researcher B, who happened to collect the data from entire population, because it turns out that it's feasible, it's like $$N=100$$ people, but researcher A didn't know that. You can think of some rare genetic diseases. Not that many people have them, so it's quite possible to get the data from all patients suffering from the condition - at least in some thought experiments like this.

The researcher B finally gets $$\bar{x_B} = 18.82$$. Obviously, $$\bar{x_A} \neq \bar{x_B}$$.

Questions:

[1] Is really the entire set of observations the population? I know that it may sound stupid (yes, it does), but I wonder if there is some catch in here? If it is, then - according to what they say - you can't use inferential statistics and you can't do eg. (statistical) significance tests when you are the researcher B. Or maybe is the population some theoretical entity and the researcher B's data is not a population?

[2] It is also said, that if you can get the data from all observations in population, you get the population's parameter, so you may write $$\mu$$ instead of $$\bar{x_B}$$. So The researcher B 's $$\bar{x_B}$$ becomes $$\mu = 18.82$$ -... right?

[3] Is the researcher A's mean $$\bar{x_A} = 17.21$$ somehow wrong?

[4] Can you point me to some valuable literature? :-) I feel I need to educate myself on this.

• $x_A$=17.21 is an estimate of $\mu$ based on (your supposedly random) sample. If you obtained another sample C, you'd obtain $x_C$ - another estimate of $\mu$. Doing so for all possible samples will give us the entire sampling distribution of the statistic (i.e. sample mean). Inference from the sample to the population is only possible because in most cases we can form reasonable expectations about what the sampling distribution of a statistic (i.e. sample mean) will look like. Commented Apr 13, 2022 at 19:19
• The key question to ask is what is the target of inference? If you want to derive conclusions about the rare disease generally, then $B$ has only a sample of the people who have had or ever will have the disease. If you only want to draw conclusions about this particular set of 100 people, then there you have it: that's your population; $B$ has a census, and $A$ has a sample. (It should be no puzzle that $A$'s mean differs from $B$'s: the first is an estimate.)
– whuber
Commented Apr 13, 2022 at 19:20

Assuming that the $$N=100$$ people that Researcher B examined is the set of people that Researcher A was interested in, then yes, that is the "population" of interest in both cases. When we refer to the "population" we are talking about whatever is the group of interest for which we want to make an inference. (For this reason, I often call it the "population of interest" to stress that meaning.) You also appear to be putting the cart before the horse on this, insofar as you are worried that Researcher B does not have an inference to make. Since Researcher B directly observes all the patients in the population, it is not really surprising that he does not require statistical inference --- statistical inference is only needed when we want to make inferences about unknown things from known things.

[1] Under the circumstances you have decribed, the $$N=100$$ values are the population of interest for both researchers. However, if Research B becomes interested in some broader theoretical group beyond those patients, then that broader group would become the new population of interest.

[2] If you are consistent/sensible in your interpretation, you can use either notation, but that is a big if. You will find that texts on model-based sampling theory will tend to reserve the Greek letters for parameters of an infinite population, whereas texts on design-based sampling theory use them for quantities of a finite population. When dealing with a finite population, I personally prefer to use the model-based notation, and therefore avoid the Greek notation for population quantities, so I would continuse to refer to the population mean as $$\bar{x}_{100}$$, not $$\mu$$. This makes it easier to extend things to make inferences about a larger infinite population if needed, and it also reduces the likelihood that someone will misinterpret the meaning of my notation.

[3] The sample mean for the smaller sample is not wrong merely by virtue of being different to the population mean. It is what it is, and so long as it was measured and recorded correctly, it is correct. Often the sample mean is used as a point estimator for the population mean, and when this occurs there will be some error in the estimator (i.e., a difference between the estimator and the thing it is trying to estimate). Estimation error is something that is expected in statistical inference, and we have teachniques to measure how big it is likely to be. The presence of estimation error does not mean that the estimator is "wrong".

[4] Big question --- I'll try to come back to this sometime.

1. You can define your population in a way that you can collect all of the observations. In the example give, one can argue that even for a rare disease you are not only interested in current people who suffer from the disease but also in people from the past and future. But for an example suppose you are interested in the cord length of cords found in my house, that is a finite number of observation which you can collect all of them.

2. Suppose you collect all of the cords in my house, and measure them taking the average, then yes that is the expected value of my house length cord distribution.

3. It is not wrong as long as he collected a random sample of the population. A negative example, taking cords specifically from my room in the house.

4. I think most introduction to statistics books have a section about sampling distribution and population.

Good questions.

1. This is debated. Under the super-population view, no. Under the finite population view, yes.

2. According to the finite population view, yes.

3. According to the finite population view, yes. According to the super-population view, researcher B still has a sample, but it's a larger and better sample than researcher A's sample.

4. See this paper and references therein.