In mathematics and statistics, a circular mean or angular mean is a mean designed for angles and similar cyclic quantities, such as daytimes, and fractional parts of real numbers. This is necessary since most of the usual means may not be appropriate on angle-like quantities. For example, the arithmetic mean of 0° and 360° is 180°, which is misleading because 360° equals 0° modulo a full cycle. 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. The circular mean is one of the simplest examples of circular statistics and of statistics of non-Euclidean spaces. 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°. Definition 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., to . 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 but generally not on 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, and there is no circular mean. (In fact, it is impossible to define a continuous mean operation on the circle.) In other words, the radius measures the concentration of the angles. 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