Suppose Bob has 50% chance to stand at each of two points $p_1, p_2$ on a unit circle. If one tries to naïvely answer the question "what is Bob's average position on the circle", an ambiguity shows up: the answer can be either the midpoint along the shortest arc between $p_{1,2}$ or along the longest.
More generally, any probability distribution for a position on the circle will seem to have an ambiguous mean. This will seem to also lead to ambiguity in the definition of the variance and higher moments.
The question is: is there a systematic way of dealing or getting rid of this ambiguity? (Note: I'm not interested in answers that assume the circle is embedded in a higher-dimensional space such as a plane, and give as an answer a point outside it. I want a point on the circle itself).