# Directional Statistics - The Fundamental Difference Between Linear and Circular Statistics

The Fundamental Difference Between Linear and Circular Statistics

A simple way to calculate the mean of a series of angles (in the interval [0°, 360°)) is to calculate the mean of the cosines and sines of each angle, and obtain the angle by calculating the inverse tangent. Consider the following three angles as an example: 10, 20, and 30 degrees. Intuitively, calculating the mean would involve adding these three angles together and dividing by 3, in this case indeed resulting in a correct mean angle of 20 degrees. By rotating this system anticlockwise through 15 degrees the three angles become 355 degrees, 5 degrees and 15 degrees. The naive mean is now 125 degrees, which is the wrong answer, as it should be 5 degrees. The vector mean can be calculated in the following way, using the mean sine and the mean cosine :

$\bar s = \frac{1}{3} \left( \sin (355^\circ) + \sin (5^\circ) + \sin (15^\circ) \right) = \frac{1}{3} \left( -0.087 + 0.087 + 0.259 \right) \approx 0.086$
$\bar c = \frac{1}{3} \left( \cos (355^\circ) + \cos (5^\circ) + \cos (15^\circ) \right) = \frac{1}{3} \left( 0.996 + 0.996 + 0.966 \right) \approx 0.986$
$\bar \theta = \left. \begin{cases} \arctan \left( \frac{\bar s}{ \bar c} \right) & \bar s > 0 ,\ \bar c > 0 \\ \arctan \left( \frac{\bar s}{ \bar c} \right) + 180^\circ & \bar c < 0 \\ \arctan \left (\frac{\bar s}{\bar c} \right)+360^\circ & \bar s <0 ,\ \bar c >0 \end{cases} \right\} = \arctan \left( \frac{0.086}{0.986} \right) = \arctan (0.087) = 5^\circ.$

This may be more succinctly stated by realizing that directional data are in fact vectors of unit length. In the case of one-dimensional data, these data points can be represented conveniently as complex numbers of unit magnitude, where is the measured angle. The mean resultant vector for the sample is then:

$\overline{\mathbf{\rho}}=\frac{1}{N}\sum_{n=1}^N z_n.$

The sample mean angle is then the argument of the mean resultant:

$\overline{\theta}=\mathrm{Arg}(\overline{\mathbf{\rho}}).$

The length of the sample mean resultant vector is:

$\overline{R}=|\overline{\mathbf{\rho}}|$

and will have a value between 0 and 1. Thus the sample mean resultant vector can be represented as:

$\overline{\mathbf{\rho}}=\overline{R}\,e^{i\overline{\theta}}.$

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