Comparing variances of vectors

For vectors we can use the covariance matrix which contains the variances per variable and the covariances.

Now I want to compare the variances of multiple point clouds. It is hard to compare the covariance matrices directly, so I would like to condense the information to a single variance scalar. (Actually its deviation, i.e. variance's square root.)

What is the typical/classical value to use?

Use the square root of the trace of the covariance matrix (the sum of its diagonal)? Or maybe not use the covariance matrix at all, e.g. computing the average Euclidean distances of each point to the mean of the point-cloud?

• Could you explain why you are comparing these variances? What information are you hoping it will convey? – whuber Nov 5 '15 at 14:17
• @whuber I want to be able to say that point cloud A varies less than B (with respect to the chosen measure of variance) – ben Nov 5 '15 at 15:32
• But that's the entire point: what aspect of this rich, multivariate complex of points are you trying to describe? One could propose all kinds of single-value measures of variation, but you haven't given us any information about how to select one that is suitable for your objectives. – whuber Nov 5 '15 at 15:34
• That interpretation would make your question overly broad and unfocused, and perhaps a subjective matter, because there are many ways to do that. Indeed, any question on this site is all about what you need to use. It's not for us to tell you why you're interested in something! – whuber Nov 5 '15 at 16:36
• There is no one "typical/classical way." Please also consult our help center, which implores you to ask "practical, answerable questions based on actual problems that you face." Requesting something abstractly "classical" would seem to have nothing to do with an actual problem. – whuber Nov 5 '15 at 20:24