Directional Statistics - Measures of Location and Spread

Various measures of location and spread may be defined for both the population and a sample drawn from that population. The most common measure of location is the circular mean. The population circular mean is simply the first moment of the distribution while the sample mean is the first moment of the sample. The sample mean will serve as an unbiased estimator of the population mean.

When data is concentrated, the median and mode may be defined by analogy to the linear case, but for more dispersed or multi-modal data, these concepts are not useful.

The most common measures of circular spread are:

• The circular variance. For the sample the circular variance is defined as:
$\overline{\mathrm{Var}(z)}=1-\overline{R}\,$
and for the population
$\mathrm{Var}(z)=1-R\,$
Both will have values between 0 and 1.
• The circular standard deviation
$S(z)=\sqrt{\ln(1/R^2)}=\sqrt{-2\ln(R)}\,$
$\overline{S}(z)=\sqrt{\ln(1/\overline{R}^2)}=\sqrt{-2\ln(\overline{R})}\,$
with values between 0 and infinity. This definition of the standard deviation (rather than the square root of the variance) is useful because for a wrapped normal distribution, it is an estimator of the standard deviation of the underlying normal distribution. It will therefore allow the circular distribution to be standardized as in the linear case, for small values of the standard deviation. This also applies to the von Mises distribution which closely approximates the wrapped normal distribution.
• The circular dispersion
$\delta=\frac{1-R_2}{2R^2}$
$\overline{\delta}=\frac{1-\overline{R}_2}{2\overline{R}^2}$
with values between 0 and infinity. This measure of spread is found useful in the statistical analysis of variance.

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