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This basic formula of finding the Standard Error (Or Standard Deviation) of a Sampling Distribution,

S.E= s/sqrt(n)

Where

s= std deviation of sample

n=sample size

Is this only for the Sampling Distributions of "Means" alone and of those that are Normally Distributed? Or does it apply to other point estimates, or other distributions?

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It is the standard deviation of the sample mean for any distribution, not just the normal. This follows from the theory of cumulants. Also, it is exact for any $n$, not just asymptotic.

It is not the standard deviation of any sample moment or other sample aggregate quantity, just of the mean.

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    $\begingroup$ It will be exact or asymptotic depending on the exact definition of s - it is not clear from the question whether S is based on the sample values (in which case it will be asymptotic) or is the population standard variation (in which case it is exact). $\endgroup$ – Zahava Kor Jun 28 '17 at 16:57

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