# Standard deviation of a cluster

I have a list of (x,y) pairs that form a cluster of points. I was able to find the mean of the cluster, but I'd like to be able to find the standard deviation of the cluster. How can I define the standard deviation? I defined the mean as the point (mean of x values, mean of y values).

What type of standard deviation value is typical for clusters like this? I'd imagine I could get a standard deviation of the spread, or something like that?

This looks good. Gaussian on a cluster:

How do I do this?

• Obtain within-cluster covariance matrix between X and Y, sum the two diagonal elements and take the root. – ttnphns Apr 6 '12 at 7:14
• Two comments: 1. Great graphic. It looks like you would have to have something like an implicit standard deviation in the clustering process, or else how would you judge points like 0.8,0.9 to belong to the red cluster, not the blue one? – Owe Jessen Apr 6 '12 at 13:04

1. One way to do it would be to count the standard deviation on $x$s and $y$s separately, to get you a box shape.
2. If you prefer the circle shape, then simply given the mean $(\bar{x},\bar{y})$ and the points $(x_i,y_i)$ compute the distance $r_i = \sqrt{(\bar{x} - x_i)^2 + (\bar{y} - y_i)^2}$ and then calculate the standard deviation of the $r_i$'s around $0$. The resulting standard deviation (or if you divide by $\sqrt{N}$ then standard error) will be the radius of the circle.