# Mean of circular quantities

In mathematics, a mean of circular quantities is a mean which is sometimes better-suited for quantities like angles, daytimes, and fractional parts of real numbers. This is necessary since most of the usual means may not be appropriate on circular quantities. For example, the arithmetic mean of 0° and 360° is 180°, which is misleading because for most purposes 360° is the same thing as 0°.[1] As another example, the "average time" between 11 PM and 1 AM is either midnight or noon, depending on whether the two times are part of a single night or part of a single calendar day. This is one of the simplest examples of statistics of non-Euclidean spaces.

## Mean of angles

Since the arithmetic mean is not always appropriate 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 within the unit disk. 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, and there is no circular mean. In other words, the radius measures the concentration of the angles.

Given the angles $\alpha_1,\dots,\alpha_n$ a common formula of the mean is

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

using the atan2 variant of the arctangent function, or

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

using complex numbers.

This computation produces a different result than the arithmetic mean, with the difference being greater when the angles are widely distributed. For example, the arithmetic mean of the three angles 0°, 0° and 90° is (0+0+90)/3 = 30°, but the vector mean is 26.565°. Moreover, with the arithmetic mean the circular variance is only defined ±180°.

## Properties

The mean $\bar{\alpha}$

$\bar{\alpha} = \underset{\beta}{\operatorname{argmin}} \sum_{j=1}^n d(\alpha_j,\beta)$, where $d(\varphi,\beta) = 1-\cos(\varphi-\beta).$
The distance $d(\varphi,\beta)$ is equal to half the squared Euclidean distance between the two points on the unit circle associated with $\varphi$ and $\beta$.