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I have been wondering about a theoretical question regarding statistics. So if I understand it correctly, in the frequentist tradition we want to make inferences from a sample to a population (of infinite size).

Now: there definetely are some populations that are smaller than others. Making an inference about all women is necessarily a subset of all humans. So sampling let's say 100 datapoints to make inferences about women should be different from sampling 100 datapoints to make inferences about humans. But whether we are making an inference about the population of female or humans in general doesn't matter in this statistical framework, is this correct? But isn't that illogical? It seems to me more robust with x datapoints to be saying something about a smaller set than a greater set - but maybe my intuition is wrong.

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Two points here. First, as regards an "infinite" population, you are really referring to a data generating process, one which can produce an infinite number of potential (think "future" if it helps to conceptualize) female responses, and one which can produce an infinite number of human responses. In this setting, the fact that female potential responses are a subset of human potential responses does not make any difference, since they are both infinite. To put it another way 100/infinity(females) = 100/infinity(humans) = 0. Here, the inference is toward the values of potential data, not toward the values of data in a pre-existing population at an instantaneous predetermined time.

Second, if the interest really is in a pristine, finite population setting, as in all females at a high school at some instantaneous time point, vs. all humans at the same instantaneous time point, and if you are able to draw pure random samples at said time point and take measurements, AND if (say) the number of females in the school was 200 and the number of humans was 400, then yes, in this setting you would have more accuracy for the female estimate. (See "finite population correction factor.") Here the inference is toward the specific population of data values at that predetermined time point.

However, this improved accuracy (given by the finite population correction) rapidly diminishes as the population size increases, so even if the interest is in the predetermined time point of values in the well-defined population, rather than in the potential values of the data-generating process, the sample size issue is for all intents and purposes negligible.

Perhaps a helpful analogy is this: Take a large pot of soup, well-stirred. Taste it using a small spoon. Now, pour half of the large pot into a different pot (the subset). Taste it again. Now, does a taste from a single spoonful provide better information when sampled from the subset?

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  • $\begingroup$ Ah that makes a lot of sense, thank you so much! I was also wondering about this partly because I was reading a study about the effects of going to space on the brain. They of course only had a very small sample, but still used the method of thinking about infinite set sizes (whereas probably not many people will ever have the chance to go to space). Would using such a statistical method bias the inferences that are made? Is it inappropriate in this case? $\endgroup$
    – Marvinsky
    May 27, 2021 at 17:13
  • $\begingroup$ OK, correction: I guess they wanted to make inferences about all cosmonauts who could go to space, so I guess that in that scenario the infinite assumption is valid. $\endgroup$
    – Marvinsky
    May 27, 2021 at 17:17
  • $\begingroup$ Yes, usually, science is interested in the broader generalities. $\endgroup$ May 28, 2021 at 14:55

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