I have a continuous random variable over a circle, so that the probability density function $\rho(\theta) = \rho(\theta+2\pi)$. I should be able to calculate its expected value and standard deviation, although I may need to define them slightly differently in order to achieve the same intuitive goals. The properties I am looking for are:
The circle statistics should reduce to regular flat-space statistics whenever my PDF is highly concentrated in one part of the circle (because curves locally appear like lines).
They should transform sensibly under a coordinate shift $\rho(\theta) \rightarrow \rho'(\theta) =\rho(\theta+\phi)$, carrying $E[\rho]\rightarrow E[\rho']=E[\rho]+\phi$ and Var$[\rho]\rightarrow $Var$[\rho']=$Var$[\rho]$. We desire these properties because the angle is an arbitrary coordinate and choice of coordinate system shouldn't determine anything significant about our answers.
How should the expected value and variance be defined so that they satisfy these properties?