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I'm trying to estimate the standard error of the mean for measurements sampled from an unknown distribution. Wikipedia says: The standard error of the mean (SEM) can be expressed as:

$$ \bar{\sigma}_x = \frac {\sigma }{\sqrt {n}}$$

where

σ is the standard deviation of the population.
n is the size (number of observations) of the sample

I was wondering if this results holds for all distributions of our random variable $X$.

In the following post, it is implied that for estimates of the standard error in higher moment (variance, skewness), the estimate is/might be dependent on the distribution:

What is the standard error of the mean of an exponential distribution of the form $Ae^{Bx}$ with N measurements?.

Specifically, when can we use: $$SE_{\bar{σ}^2} = σ^2 \sqrt{\frac{2}{n-1}}$$

Is there a certain textbook/easy paper you recommend for these kind of questions?

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