Does standard error for a sample mean make assumptions about the distribution?

Question

Suppose I calculate standard deviation of a sample mean as

$ \sigma / \sqrt{n} $

where $\sigma$ is the population standard deviation and $n$ the sample size.

Have I just made an assumption that the population has normal distribution?

If so, how would this formula differ if the population distribution was exponential $e^{-x}$? Intuitively it would seem wrong as our sample might have missed a very large value on one side of the distribution only, whereas a normal distribution would have outliers on both sides?

Answer 1

Under the usual assumptions (ie. i.i.d. samples), the sample mean will have standard deviation $\dfrac{\sigma}{\sqrt{n}}$ regardless of the population distribution, because the Bienaymé formula is distribution-free. The assumption that the population is normal is only required for the distribution of the sample mean to be also exactly normally distributed.