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I keep seeing sources stating, without proof, that the standard deviation of the sampling distribution of the sample mean:

$$\sigma/\sqrt{n}$$

is an approximate formula that only holds if the population size is at least 20 times the sample size.

Can anyone offer a proof of this statement, or disprove it? If it's false, could you please explain the intuition behind why anyone would come up with this in the first place?

Here are two places in which I've seen this claim:

1) "You often see this "approximate" formula in introductory statistics texts. As a general rule, it is safe to use the approximate formula when the sample size is no bigger than 1/20 of the population size." - https://stattrek.com/sampling/sampling-distribution.aspx

2) "the formula for the standard deviation of the sampling distribution of the sample mean, $\sigma/\sqrt{n}$, holds approximately if the population is finite and much larger than (say, at least 20 times) the size of the sample". A business statistics textbook.

EDIT:

Yes, this is in the context of sampling from finite populations.

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    $\begingroup$ That must be in the context of sampling from a finite population! For most applications, that is irrelevant. $\endgroup$ – kjetil b halvorsen Apr 5 '20 at 16:18
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    $\begingroup$ Thank you, but even if it's irrelevant in most applications, there are some contexts in which it would be relevant. I'm still interested in knowing the "why" behind this. $\endgroup$ – StatsSorceress Apr 5 '20 at 16:39
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What you are told is only relevant when you are sampling from a finite population of size $N$, with simple random sampling without replacement. For most applications, there is no definite finite population, so what you are told is irrelevant. It is also irrelevant when you are samling with replacement.

When relevant, there is a finite population correction explained here: Explanation of finite correction factor and here for more details, a web pdf.

But, even if you think you have a finite population, it might not be relevant. Most uses of statistics are analytic, not enumerative. So if you are sampling from this years hospital patients, presumably a finite population, presumably you are not only interested in that specific population, but want to generalize to a larger population from which that one was drawn, from which next year patients will be drawn, ... and then the finite population aspects are irrelevant.

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    $\begingroup$ Thank you. To be pedantic for a moment: this 2020's hospital patients are naturally going to be very different from 2019's due to the pandemic. So I wouldn't want to generalize! In this (admittedly rare) exception, I would consider the population finite. $\endgroup$ – StatsSorceress Apr 5 '20 at 17:35
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    $\begingroup$ Would it be fair to say that if you're sampling with replacement, you have an artificially infinite population? $\endgroup$ – StatsSorceress Apr 6 '20 at 15:02
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It is only a rule of thumb that, when sample size reaches 5% of population size, then the finite population correction becomes relevant. Nothing specifically changes at that point. One primary use for this rule is for a researchers attempting to choose sample size for a study. Given a certain cost per respondent, you would tend to see a noticeable difference in total cost--naive estimate vs corrected estimate--at that point. Obviously, for some data collection methods, cost is very low, and so the difference in cost would also be low.

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  • $\begingroup$ Interesting point, Ed, thanks. Any idea where this rule of thumb came from? $\endgroup$ – StatsSorceress Apr 5 '20 at 17:36
  • $\begingroup$ You will find it in marketing research research textbooks dating to the 1980s. Finite population corrections for statistics other than means and proportions are somewhat hard to find and not widely cited. $\endgroup$ – Ed Rigdon Apr 5 '20 at 22:37

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