Are there pre-existing and widely used measures for the level of "hierarchy" in a data set? Learning about hierarchical clustering, I was wondering if there is a commonly used measure for the amount of hierarchy in a data set?
What I mean is: If the data points are all uncorrelated normal random variables with the same mean and std, clearly there is no hierarchy and thus a hierarchy measure should be 0, or close to 0.
If, on the other hand, there is a strong hierarchical relationship going down many levels, then the measure should be 1 (normalized) or something else far from 0.
EDIT: I should add that I can quite readily come up with a number of ways to approach this question, but wanted to figure out if there's already a widely-used / accepted / standard approach to this, and somehow googling for "hierarchy measure" or something along those lines doesn't yield useful results.
 A: Well, I haven't heard back with any definite answer, so I thought I'd respond with what I ended up using:
I take my data set for which I want to check the level of hierarchy, and I compute a linkage matrix for it (using scipy.cluster.hierarchy.linkage) from the scipy clustering package. 
This gives me a clustering tree expressed in that linkage matrix, and for that tree I compute the cophenetic correlation coefficient.
This coefficient measures the correlation between the distance of two elements and the height of the node in the clustering tree (dendrogram) at which the two elements first get joined together.
If you have a high coefficient (close to 1), that means elements that are close together meet early during the agglomerative clustering, and elements that are far apart meet later. 
Now, there's a bunch of knobs you can turn, such as what metric to use, and what sort of linkage (single, complete, ward, ...), but I found that it works okay for a ballpark estimate of "yup, this data set has lots of hierarchy and, nope, this data set doesn't".
Early experiments with the silhouette coefficient, btw, proved utterly disappointing.
