Mean of circular quantities

In mathematics, a mean of circular quantities is a mean which is suited for quantities like angles, daytimes, and fractional parts of real numbers. This is necessary since most of the usual means fail on circular quantities. For example, the arithmetic mean of 0° and 360° is 180°, although 0° would be clearly the better choice.

Mean of angles
Since the arithmetic mean is not effective for angles, the following method can be used to obtain both a mean value and measure for the variance of the angles:

Convert all angles to corresponding points on the unit circle, e.g. $$\alpha$$ to $$(\cos\alpha,\sin\alpha)$$. That is convert polar coordinates to Cartesian coordinates. Then compute the arithmetic mean of these points. The resulting point will lie in the unit circle. Convert that point back to polar coordinates. The angle is a reasonable mean of the input angles. The resulting radius will be 1 if all angles are equal. If the angles are uniformly distributed on the circle, then the resulting radius will be 0. In other words, the radius measures the concentration of the angles.

Given the angles $$\alpha_1,\dots,\alpha_n$$ the mean is computed by


 * $$M \alpha = \operatorname{atan2}\left(\frac{1}{n}\cdot\sum_{j=1}^n \sin\alpha_j, \frac{1}{n}\cdot\sum_{j=1}^n \cos\alpha_j\right) $$

using the atan2 variant of the arctangent function, or


 * $$M \alpha = \arg\left(\frac{1}{n}\cdot\sum_{j=1}^n \exp(i\cdot\alpha_j)\right) $$

using complex numbers.

Properties
The mean $$M\alpha$$
 * maximizes the likelihood of the mean parameter of the circular normal distribution and
 * minimizes the sum of a certain distance on the circle, more precisely $$M\alpha = \underset{\beta}{\operatorname{argmin}} \sum_{j=1}^n d(\alpha_j,\beta)$$, where $$d(\varphi,\beta) = 1-\cos(\varphi-\beta)$$. The distance $$d(\varphi,\beta)$$ measures the length of the projection of the segment between the two points on the unit circle associated with $$\varphi$$ and $$\beta$$ onto the line through the origin and the point associated with $$\beta$$.