Editing Arithmetic mean (section)

===Angles===
{{main|Mean of circular quantities}}

Particular care is needed when using cyclic data, such as phases or [[angle]]s. Taking the arithmetic mean of 1° and 359° yields a result of 180[[degree (angle)|°]].
This is incorrect for two reasons:
*Firstly, angle measurements are only defined up to an additive constant of 360° (<math>2\pi</math> or <math>\tau</math>, if measuring in [[radian]]s). Thus, these could easily be called 1° and -1°, or 361° and 719°, since each one of them produces a different average.
*Secondly, in this situation, 0° (or 360°) is geometrically a better ''average'' value: there is lower [[Statistical dispersion|dispersion]] about it (the points are both 1° from it and 179° from 180°, the putative average).

In general application, such an oversight will lead to the average value artificially moving towards the middle of the numerical range. A solution to this problem is to use the optimization formulation (that is, define the mean as the central point: the point about which one has the lowest dispersion) and redefine the difference as a modular distance (i.e., the distance on the circle: so the modular distance between 1° and 359° is 2°, not 358°).

{{AM_GM_inequality_visual_proof.svg}}