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:

enter image description here

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?

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    $\begingroup$ 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. $\endgroup$ – TonyK Jun 28 '14 at 10:18
  • $\begingroup$ @TonyK Standard deviation of the points. $\endgroup$ – padawan Jun 28 '14 at 10:35
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    $\begingroup$ 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? $\endgroup$ – TonyK Jun 28 '14 at 12:04
  • $\begingroup$ @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 $\endgroup$ – padawan Jun 28 '14 at 12:34

I don't understand your gripe with standard deviation. However, here's something that should work:

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.


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