Measure of spread of a multivariate normal distribution What is a good measure of spread for a multivariate normal distribution?
I was thinking about using an average of the component standard deviations; perhaps the trace of the covariance matrix divided by the number of dimensions, or a version of that. Is that any good?
Thanks
 A: I would go with either trace or determinant with a preference towards trace depending on the application. They're both good in that they're invariant to representation and have clear geometric meanings. 
I think there is a good argument to be made for Trace over Determinant. 
The determinant effectively measures the volume of the uncertainty ellipsoid. If there is any redundancy in your system however then the covariance will be near-singular (the ellipsoid is very thin in one direction) and then the determinant/volume will be near-zero even if there is a lot of uncertainty/spread in the other directions. In a moderate to high-dimensional setting this occurs very frequently
The trace is geometrically the sum of the lengths of the axes and is more robust to this sort of situation. It will have a non-zero value even if some of the directions are certain. 
Additionally, the trace is generally much easier to compute. 
A: What about the determinant of the sample variance-covariance matrix: a measure of the
squared volume enclosed by the matrix within the space of dimension of the measurement vector. Also, an often used scale invariant version of that measure is the determinant of the sample correlation matrix: the volume of the space occupied within the dimensions of the measurement vector. 
A: Another (closely related) quantity is the entropy of the distribution: for a multivariate Gaussian this is the log of the determinant of the covariance matrix, or
$\frac{1}{2} \log |(2\pi e)\Lambda|$
where $\Lambda$ is the covariance matrix.  The advantage of this choice is that it can be compared to the "spread" of points under other (e.g., non-Gaussian) distributions.  
(If we want to get technical, this is the differential entropy of a Gaussian). 
