Circular statistics

Circular statistics is the subdiscipline of statistics that deals with circular data. The fact that 0 degrees and 360 degrees are identical angles, so that for example 180 degrees is not a sensible mean of 2 degrees and 358 degrees, provides one illustration that special statistical methods are required for the analysis of circular data.

Other examples of data that may be regarded as circular include statistics involving days of the week, months of the year, compass directions, and so on.

The fundamental insight is that such data are often best handled not as numbers, but as unit vectors. So to average a number of times of the day, treat each time as a unit vector whose angle is the appropriate fraction of a circle, compute their sum, and divide by N to get a mean with both direction and magnitude. The closer the times are to being completely random, the smaller that vector mean will be, whereas if the mean has a large magnitude that would imply a significant tendency in the data.

The equivalent in circular statistics of the Gaussian or Normal distribution in conventional statistics is the von Mises distribution.