Editing Directional statistics (section)

== Moments ==
The raw vector (or trigonometric) moments of a circular distribution are defined as

:<math>
m_n=\operatorname E(z^n)=\int_\Gamma P(\theta) z^n \, d\theta
</math>

where <math>\Gamma</math> is any interval of length <math>2\pi</math>, <math>P(\theta)</math> is the [[Probability density function|PDF]] of the circular distribution, and <math>z=e^{i \theta}</math>. Since the integral <math>P(\theta)</math> is unity, and the integration interval is finite, it follows that the moments of any circular distribution are always finite and well defined.

Sample moments are analogously defined:

:<math>
\overline{m}_n=\frac{1}{N}\sum_{i=1}^N z_i^n.
</math>

The population resultant vector, length, and mean angle are defined in analogy with the corresponding sample parameters.

:<math>
\rho=m_1
</math>

:<math>
R=|m_1|
</math>

:<math>
\theta_n=\operatorname{Arg}(m_n).
</math>

In addition, the lengths of the higher moments are defined as:

:<math>
R_n=|m_n|
</math>

while the angular parts of the higher moments are just <math>(n \theta_n) \bmod 2\pi</math>. The lengths of all moments will lie between 0 and 1.