# How to measure points scattering in a circle

I have circles containing points (x,y). I would like to measure the scattering of the points within the circle.
For example, in the following picture, circle A will have a higher value since the points are much more scattered across the circle.

Notice that the circles have varying value - so we can have a circles with different radiuses.
For example in the following picture, although the points are the same - circle C will have a higher value because the points are scattered across the whole circle.

Do you know a measurement which I can use for such purpose?
Thanks!

• Wouldn't mean-square distance from the points' center of mass, divided by the radius of the circle, do the trick? – gmvh Sep 23 at 11:10

Judging from your pictures, it should be sufficient to consider the root-mean-square distance $$\rho$$ of the points $$\vec{x}_k$$ from their center of mass $$\vec{\mu}$$, divided by the radius $$R$$ of the circle: \begin{align} \vec{\mu} &= \frac{1}{N}\sum_{k=1}^N\vec{x}_k, \\ \rho &= \sqrt{\left(\frac{1}{N}\sum_{k=1}^N\left\lVert\vec{x}_k-\vec{\mu}\right\rVert^2\right)}, \\ S &=\frac{\rho}{R}, \end{align} where $$S$$ is your measure of scattering within the circle.