Editing Directional statistics (section)

== Measures of location and spread ==
Various measures of [[central tendency]] and [[statistical dispersion]] may be defined for both the population and a sample drawn from that population.{{sfn|Fisher|1993}}

=== Central tendency ===
{{Further|Circular mean}}

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 (statistics)|mode]] may be defined by analogy to the linear case, but for more dispersed or multi-modal data, these concepts are not useful.

=== Dispersion ===
{{See also|Yamartino method}}

The most common measures of circular spread are:

* The '''{{visible anchor|circular variance|Variance}}'''. For the sample the circular variance is defined as: <math display="block">
\overline{\operatorname{Var}(z)} = 1 - \overline{R}
</math> and for the population <math display="block">
\operatorname{Var}(z) = 1 - R
</math> Both will have values between 0 and 1.
* The '''{{visible anchor|circular standard deviation|Standard deviation}}''' <math display="block">
S(z) = \sqrt{\ln(1/R^2)} = \sqrt{-2\ln(R)}
</math> <math display="block">
\overline{S}(z) = \sqrt{\ln(1/{\overline{R}}^2)} = \sqrt{-2\ln({\overline{R}})}
</math> 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. Note that for small <math>S(z)</math>, we have <math>S(z)^2 = 2 \operatorname{Var}(z)</math>.
* The '''{{visible anchor|circular dispersion|Dispersion}}''' <math display="block">\delta = \frac{1-R_2}{2R^2}</math> <math display="block">
\overline{\delta}=\frac{1-{\overline{R}_2}}{2{\overline{R}}^2}
</math> with values between 0 and infinity. This measure of spread is found useful in the statistical analysis of variance.