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In mathematics and statistics, a circular mean or angular mean is a mean designed for angles and similar cyclic quantities, such as times of day, and fractional parts of real numbers. This is necessary since most of the usual means may not be appropriate on angle-like quantities.
In probability and statistics, a circular distribution or polar distribution is a probability distribution of a random variable whose values are angles, usually taken to be in the range [0, 2π). [1] A circular distribution is often a continuous probability distribution , and hence has a probability density , but such distributions can also be ...
Directional statistics (also circular statistics or spherical statistics) is the subdiscipline of statistics that deals with directions (unit vectors in Euclidean space, R n), axes (lines through the origin in R n) or rotations in R n.
If we assume that the mean lies in the interval [,], then Arg (¯) will be a (biased) estimator of the mean . Viewing the z n {\displaystyle z_{n}} as a set of vectors in the complex plane, the R ¯ 2 {\displaystyle {\bar {R}}^{2}} statistic is the square of the length of the averaged vector:
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A 10,000 point Monte Carlo simulation of the distribution of the sample mean of a circular uniform distribution for N = 3 Probability densities (¯) for small values of . Densities for N > 3 {\displaystyle N>3} are normalised to the maximum density, those for N = 1 {\displaystyle N=1} and 2 {\displaystyle 2} are scaled to aid visibility.
The circular standard deviation, which is a useful measure of dispersion for the wrapped normal distribution and its close relative, the von Mises distribution is given by: s = ln ( R − 2 ) 1 / 2 = σ {\displaystyle s=\ln(R^{-2})^{1/2}=\sigma }
1 Circular mean quirkiness and other random thoughts. 2 comments. ... 7 N-dimensional version, and quaternion version. Toggle the table of contents. Talk: Circular mean.