Talk:Mean of circular quantities

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[edit] Circular mean quirkiness and other random thoughts

There are some weird quirks to this calculation of mean that probably ought to be noted.

  1. if the radius calculated from all unit vectors is zero, the is no circular mean (noted)
  2. perhaps some notability of the certainty of the angle retrieved as it relates to the radius.... or if there's multi-modal distributions and they average out to have a radius near zero, the resulting calculated circular mean could be all over the place and not anywhere near the "true" angle one would hope to recover from statistics (although we could be dead certain that the "true radius" estimated by the sum of all those vectors would be quite small). The concept of computational stability is involved here as well, although I wouldn't know what article to link to for further guidance.
  3. Just for my curiosities sake: Is there a physical analog to this mean? What is it? I can't think of one off the top of my head but I'm certain it exists. (actually, I do have some ideas considering an ideal disk or sphere that I think work, but I'm working now and can't really work on the idea further... I guess this is Original Research, anyways, so unless noted in a book somewhere, probably not include-able. But it also relates to idea number 2.)

Oh yes, I know.. balancing a wheel on the tire, except the concept there is to not have a circular average(! having one would be a bad thing!!)

I'm not sure if any of the above makes sense but hopefully it does. Root4(one) 21:36, 19 November 2007 (UTC)


Some other random thoughts/notes:
- For the physical analog, as you mentioned, it can be seen as the angle defined by the center of mass of the wheel.
- The method could be generalized to a weighted mean (in order to unbalance the wheel :)
- A tricky question: what does "reasonable mean" means? knowing that

- the arithmetic mean can be calculated by taking the angle minimizing the variance of the angles, and you then obtain a slightly different mean than this reasonable mean
- if you restrict the angles to [-90° 90°], and compare it with the ordinary arithmetic mean (which then makes sense, doesn't it?), the 2 means are again slightly different

The "reasonable" (or "slightly") should maybe be detailed for more accuracy, but I didn't manage to get anything consistent myself... Kiwux (talk) 14:18, 9 April 2008 (UTC)