Standard deviation for a half-normal distribution I have a vector of numbers which is generated from a normal distribution with standard deviation $\sigma$ and $\mu = 0$. When I'm turning these numbers into absolute values, I suppose that the they will follow half-normal distribution with the same standard deviation.
My question is, if I initially start at having a vector of absolute values which generated from half-normal distribution, would it be mathematically sound to add the negative values of these numbers and then find the combined standard deviation between absolute values and negative values to recreate the normal distribution?

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*The half normal distribution is usually parametrised so that the $\sigma$ from the corresponding normal distribution is its scale parameter. However, this is not the standard deviation of it, which is $\sigma\sqrt{1-\frac{2}{\pi}}$, see https://en.wikipedia.org/wiki/Half-normal_distribution. It should be intuitive that the sd is smaller than $\sigma$; there is more variance if observations can deviate on both sides of zero - cutting one half off a region where no observations is left and putting them in the place where the other observations are lowers variation.


*Regarding your question: If you want a proper random sample from the normal distribution, this is not correct, because although the normal distribution is symmetric, samples of it are not perfectly symmetric. Particularly the probability is zero that if you have $x$ in a normal sample you also have $-x$ in it. The number of
observations left and right from 0 can be the same, but more often than not, due to random variation, it won't be the same.
