text: Given the angles  a common formula of the mean using the atan2 variant of the arctangent function is
  Using complex arithmetic
  An equivalent definition can be formulated using complex numbers:
  .
  In order to match the above derivation using arithmetic means of points, the sums would have to be divided by . However, the scaling does not matter for  and , thus it can be omitted.
  This may be more succinctly stated 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 .