# Variance and CDF for non-directional wrapped gaussian distribution (bounded angles [0, pi])

How can we calculate the standard error sigma and the chance of a sample being within N sigma (CDF) for values bound to [0, pi] (absolute angular errors)?

You can perhaps think of a blindfolded and/or drunken archer shooting at a target on the wall of a circular room. Good archers will get a tight distribution around the target, but that distribution will be not be strictly Gaussian when the values approach the limits [-pi, pi] creating a wrapped Gaussian distribution.

Yet a subset of that problem (with additional practical applications) is when is also not possible to measure the direction of the error, and therefore the space is additionally limited to [0, pi] (the absolute angular error) In this case, the regular average value is not the expected value of the distribution, which should be still, at least close to 0