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We know the variane of the distribution of the sample mean is $\sigma_{m}^2=\sigma^2/n$, where $n$ is the sample size and $\sigma^2$ is the population variance.

Say we have two populations -- all male in your univeristy department (Po1, 120 males) and all male in your city (Po2, 120K males).

Assuming the population variance of Po1 is equal to Po2, and we fix the sample size $n=100$ for both populations. Then by $\sigma_{m}^2=\sigma^2/n$, the two variances of sample mean are also identical.

But this is quite counter-intutive, isn't it? Since the sample size $n$ is much closer to the size of Po1 than Po2, but still they have the same varaince in thoese point estimates. In other words, the sample size should be somehow determined by the population size -- the larger the population size the more samples you need to have a good estimate.

Is there any intutive reasons behind this or is it simply becasue the assumption ``population variance of Po1 is equal to Po2'' is counter-intutive? Thanks.

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If your sample is without replacement from a finite population size $N$, then the variance of the sample mean is not $\dfrac{\sigma^2}{n}$ but $\dfrac{N-n}{N-1}\dfrac{\sigma^2}{n}$

So in your example of $n=100$, $N_1=120$ and $N_2=120000$, then instead of a variance of $\dfrac{\sigma^2}{100}$, we would get $\dfrac{\sigma^2}{595}$ for the smaller population and about $\dfrac{\sigma^2}{100.083}$ for the variance population, with the variance for the smaller population substantially smaller, in line with your intuition

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  • $\begingroup$ Thank you, this is a good answer! But what if my sampling is with replacement? $\endgroup$ May 4, 2021 at 9:32
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    $\begingroup$ @weidade3721 With replacement, you get back to the variance of the sample mean is being $\frac{\sigma}{n}$. This also makes intuitive sense: you would get the same distribution for the sample mean with replacement from a population with $120$ as from a population with $1000$ copies of those $120$ $\endgroup$
    – Henry
    May 4, 2021 at 9:36
  • $\begingroup$ Thanks for the reply.. But why ``the sample size is determined by the population size -- the larger the population size the more samples you need to have a good estimate.'' like the eqution here surveymonkey.co.uk/mp/sample-size-calculator $\endgroup$ May 4, 2021 at 9:39
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    $\begingroup$ @weidade3721 $\frac{\sigma^2}{n}$ is a decreasing function of $n$. If you want a small variance for the sample mean then you need a large sample size $n$ $\endgroup$
    – Henry
    May 4, 2021 at 9:46

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