Directional Statistics - Distribution of The Mean

Distribution of The Mean

Given a set of N measurements the mean value of z is defined as:

\overline{z}=\frac{1}{N}\sum_{n=1}^N z_n

which may be expressed as

\overline{z} = \overline{C}+i\overline{S}


\overline{C} = \frac{1}{N}\sum_{n=1}^N \cos(\theta_n) \text{ and } \overline{S} = \frac{1}{N}\sum_{n=1}^N \sin(\theta_n)

or, alternatively as:

\overline{z} = \overline{R}e^{i\overline{\theta}}


\overline{R} = \sqrt{\overline{C}^2+\overline{S}^2}\,\,\,\mathrm{and}\,\,\,\,\overline{\theta} = \mathrm{ArcTan}(\overline{S},\overline{C}).

The distribution of the mean for a circular pdf P(θ) will be given by:

P(\overline{C},\overline{S}) \, d\overline{C} \, d\overline{S} =
P(\overline{R},\overline{\theta}) \, d\overline{R} \, d\overline{\theta} =
\int_\Gamma ... \int_\Gamma \prod_{n=1}^N \left

where is over any interval of length and the integral is subject to the constraint that and are constant, or, alternatively, that and are constant.

The calculation of the distribution of the mean for most circular distributions is not analytically possible, and in order to carry out an analysis of variance, numerical or mathematical approximations are needed.

The central limit theorem may be applied to the distribution of the sample means. (main article: Central limit theorem for directional statistics). It can be shown that the distribution of approaches a bivariate normal distribution in the limit of large sample size.

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