In biomolecular science, a dihedral angle is periodic with a period of $2\pi$. It ranges from -180 to 180 degrees. Now, if for 5 dihedral angles, I want to calculate their mean and variance. Will the mean and variance also be periodic? If so, what would be the periods? Also, what would be the correct way to calculate the mean and standard deviation that take the periodicity into account?
The answer I have in my mind is that the mean will be periodic with a period of $2\pi$ as well but the variance is not periodic because it can not be seen as a vector, but I'm not absolutely sure ...
As for the calculation of the mean, I think the best way is probably to use the two-argument arctangent function. For example, if we have $\theta_1$ and $\theta_2$, instead of calculating the mean as $\frac{1}{2}(\theta_1 + \theta_2)$, it should be $$\operatorname{atan}\left(\frac{1}{2}\left(\sin(\theta_1) + \sin(\theta_2)\right), \frac{1}{2}\left(\cos(\theta_1) + \cos(\theta_2)\right)\right)$$
Namely, the two arguments of the function atan is the average of the sine and cosine functions. However, I'm not so sure about the correct way to calculate the variance. I don't think it is as simple as using the atan function, with the two arguments being the variance of sine and cosine functions because that would lead to a variance of 90 between -180 and +180, while it should have been 0.