# Is there an unambiguous way to define the average (and higher moments) for a probability density on a circle?

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).

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IIRC it's usually always the "small arc" that is considered... – J. M. May 6 '11 at 3:43
@ J.M. Yes, for two points that would make sense - but the question is what to do with a generic probability distribution? – David May 6 '11 at 3:53
There exists something called "directional statistics": en.wikipedia.org/wiki/Directional_statistics . – Michael Lugo May 6 '11 at 4:43
@Michael: ...which exactly assume(s) the circle is embedded in a higher-dimensional space and give(s) as an answer a point outside it (as is, to me, the natural way to go). – Did May 6 '11 at 7:27
@Didier: basically all I know about directional statistics is that they have a Wikipedia article. – Michael Lugo May 6 '11 at 15:17

It depends on what you mean by "systematic". You can arbitrarily place a cut somewhere and parametrize the circle using angles $\phi\in[0,2\pi]$. The moments of these angles are unambiguous, but also arbitrary. There can't be a non-arbitrary way to do this because of the symmetry of the circle. If you have two points opposite each other, or more generally $n$ points at angles $2\pi/n$, there's no reason to pick any particular point as the average.

If you have additional structure on the circle, i.e. the circle is $U(1)$ or $\{z\in\mathbb C\mid \lvert z\rvert=1\}$, then you can use that for a less arbitrary definition of angles measured from the identity, but I doubt that the result will be particularly useful.

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Yes, of course there must be some arbitrariness, but the question is 'how arbitrary'. When there are only 2 possible positions, there are also 2 possible average points. For a uniform distribution over the entire circle, any point can be the average. What happens for a more generl distributions? Is there in general a different average point for every different reference 'zero' angle? – David May 6 '11 at 12:58
@David: It sounds like this may be what you're looking for? en.wikipedia.org/wiki/Mean_of_circular_quantities – joriki May 6 '11 at 13:05
yes it does sound like this. Thanks! – David May 7 '11 at 14:12

If you wish to have a specific, unambiguous "average position" for any distribution then, in particular, you expect to have one for the uniform distribution. This implies that whatever your definition is, it must give preference to some point on the circle.

Granting that, you can parametrize the circle by the angle $\theta$ relative to some fixed radius. With that, the average and all other moments are unambiguously defined as $\int \theta^k dp(\theta)$. Of course this definition breaks the symmetry of the circle but, as we've seen, this is unavoidable.

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Shouldn't the random variable with measure $p$ be $\theta$ and not $x$ in your integral? Also, I assume $p$ is a distribution and not a density? – JasonMond May 6 '11 at 5:48
sure, corrected. I didn't define $p$ either way, assumed it was clear from context. – Alon Amit May 6 '11 at 14:52