I found here the formula for computing the sample size $n$ of a finite population $N$ $$ n = \frac{n_\infty}{1 + \frac{n_\infty - 1}{N}} $$ where the sample size for an infinite population $n_\infty$ is given as $$ n_\infty = \frac{z^2 p(1-p)}{c^2} $$ with the the z-score $z$, the confidence interval $c$ and $p$ being the proportion of the population picking a specific choice. However I found no derivation of this formula. Can someone give a mathematical derivation of this formula?

  • $\begingroup$ The proof is a few pages. See Cochran, 1973, page 73. $\endgroup$ Commented Jan 24, 2016 at 9:21
  • $\begingroup$ @StatsStudent Do you have a link to some online resource with the proof, too? $\endgroup$
    – asmaier
    Commented Jan 24, 2016 at 9:30
  • $\begingroup$ Sorry. I don't. The Cochran books is considered the "bible" of sampling. You should find it in any academic library. $\endgroup$ Commented Jan 24, 2016 at 9:42
  • $\begingroup$ link for Cochran 1977: archive.org/download/Cochran1977SamplingTechniques_201703/… $\endgroup$
    – Kushdesh
    Commented Jan 24, 2019 at 9:30

1 Answer 1


Amazingly a pretty complete derivation was given at math.stackexchange.com at https://math.stackexchange.com/a/1357604/27609.

Another derivation can be found online here.


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