# Finding how “spreaded” a point cloud in 3D

I don't know the proper term for "spreaded" but what I want to find is, a value that indicates how far is an average point from the centroid.

I think this is standard deviation of the point set, but I need metrics.

For instance, consider the figures:

Here, blue points are centroids and black points are the points in the cloud.
In both cases, standard deviation is the same, but the first cloud is more "spreaded".

If a spread factor $\alpha$ were to be given, how would you compute $\alpha$ given the 3D coordinates of the points?

• The standard deviation of what? Distance from the centroid? If so, the standard deviation of the first set is much more than the standard deviation of the second set. – TonyK Jun 28 '14 at 10:18
• @TonyK Standard deviation of the points. – padawan Jun 28 '14 at 10:35
• It seems to me that you must have misunderstood what standard deviation means! Please update your post with the definition that you are using: given $n$ points $(x_i,y_i)$, how do you calculate the "standard deviation of the points", as you call it? – TonyK Jun 28 '14 at 12:04
• @TonyK But I don't want to compute the st.dev. I want to compute how spreaded are the points or how cumulated they are. Sorry but I don't know the terminology – padawan Jun 28 '14 at 12:34

Let $\{(x_i,y_i,z_i): i = 1,\dots n\}$ be a collection of points. Let $(x,y,z) = \frac 1n \sum_{i=1}^n (x_i,y_i,z_i)$ be the centroid. We can take $$\sigma^2 = \frac{\sum_{i=1}^n ((x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2)}{n}$$ $\sigma = \sqrt{\sigma^2}$ gives you the standard deviation in Euclidean distance of all points from the centroid, which should be exactly what you're looking for.