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I read something about standard error, which tells that sample mean is not accurate estimation because we do not sample full population of size N. But, what if sample size n = N or exceeds N, i.e. n > N? Can standard error can be more accurate than the standard deviation?

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    $\begingroup$ The sample size cannot exceed the population size. And standard error and standard deviation aren't comparable in terms of accuracy. $\endgroup$
    – Peter Flom
    May 5, 2014 at 11:01
  • $\begingroup$ If n=N, you've either sampled some points repeatedly, in which case you need to account for that; or you've sampled the entire population, in which case it's (usually, for most purposes) of questionable value to estimate a population parameter, considering you can simply calculate it. $\endgroup$
    – jona
    May 5, 2014 at 11:11
  • $\begingroup$ @PeterFlom Ok, it seems that I started to understand what std error measures variations of the means, which does vanishes with larger sample sizes, whereas std dev measures variation of indidividuals, which is a constant for the population means. So, it could be the answer if you elaborate your second point. $\endgroup$
    – Val
    May 5, 2014 at 11:11
  • $\begingroup$ The standard error of the mean is derivable from the standard deviation and the sample size. So, to the extent that it makes sense to even ask whether one is more "accurate" than the other, the answer must be "they are equally accurate". But since they measure different things, how can one be more accurate than the other? $\endgroup$
    – Peter Flom
    May 5, 2014 at 11:17

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Assuming that we are talking about sampling without replacement, if the sample size exceeds the population size, you need to rethink your stipulated "population size". When $n=N$ in this case you have sampled the entire population (what we call a "census") and so all measurable descriptive quantities relating to the population should be known. Sensible estimators of such quantities will have zero standard error in this case.

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