by realizing that directional data are in fact vectors of unit length. In the case of one-dimensional data, these data points can be represented conveniently as complex numbers of unit magnitude , where is the measured angle. The mean resultant vector for the sample is then: The sample mean angle is then the argument of the mean resultant: The length of the sample mean resultant vector is: and will have a value between 0 and 1. Thus the sample mean resultant vector can be represented as: Similar calculations are also used to define the circular variance. Properties The circular mean maximizes the likelihood of the mean parameter of the von Mises distribution and minimizes the sum of a certain distance on the circle, more precisely The distance is equal to half the squared Euclidean distance between the two points on the unit circle associated with and . Example A simple way to calculate the mean of a series of angles (in the interval [0°, 360°)) is to calculate the mean of the cosines and sines of each angle, and obtain the angle by calculating the inverse tangent. Consider the following three angles as an example: 10, 20, and 30 degrees. Intuitively, calculating the mean would involve adding these three angles together and dividing by 3, in this case indeed resulting in a correct mean angle of 20 degrees. By rotating this system anticlockwise through 15 degrees the three angles become 355 degrees, 5 degrees and 15 degrees. The arithmetic mean is now 125 degrees, which is the wrong answer, as it should be 5 degrees. The vector mean can be calculated in the following way, using the mean sine and the mean cosine : Generalizations Spherical mean Weighted spherical mean A weighted spherical mean can be defined based on spherical linear interpolation. See also Circular distribution Circular standard deviation Directional statistics Centroid Center of mass Fréchet mean References Further reading Jammalamadaka, S. Rao and SenGupta, A. (2001). Topics in Circular Statistics, Section 1.3, World Scientific Press, Singapore. External links Circular Values Math and Statistics with C++11, A C++11 infrastructure for circular values (angles, time-of-day, etc.) mathematics and statistics Means Directional statistics