Estimation of standard error

I was doing some self study and came across the following formulae for estimating standard errors:

Formulae 1:

Formulae 2:

I understand that these two can all be used when the Population Standard Deviation is unknown. But I don't really understand why one has that additional part at the back.

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The first term is a biased estimation of the standard deviation, and the second is another estimate when the population you sample from is small, so your sample has a size comparable to the population. See details here and references therein.

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Thanks juampa. Am I right to say that the first is Biased and for generally larger population, while the second is unbiased and for small populations? – user1275515 Apr 20 '14 at 15:04
It is still biased. If you take the second estimate, and consider N to be much, much larger than n, then the factor is almost 1 and you are left with the same estimate. It is a correction to account for the fact that in that setting not all data from the finite population are observed. – jpmuc Apr 20 '14 at 15:42

Used when sample is done with replacement from finite / infinite population

Used when sample is done without replacement from finite / infinite population

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If you are interested in the derivation, you can have a look here (onlinecourses.science.psu.edu/stat414/node/264) – jpmuc Apr 20 '14 at 19:28