Measure of dispersion over unordered set I'm looking for a measure of dispersion, such as standard deviation, that can be used when distributing to an unordered set.
Specifically: A bucket distribution assigns a non-negative value to each bucket in a finite set.  The sum of all assigned values is one.  (So far, it's like a probability distribution).  However: The buckets in the set can be distinguished but have no order.  It's this which makes it different than a probability distribution.
We can represent bucket distributions as lists of decreasing non-negative numbers that sum to 1.  Eg [1] or [1/2,1/2] or [1/3,1/3,1/6,1/6] or [1/2,1/4,1/8,1/16...].
I'd like to be able to measure how dispersed a particular bucket distribution is.  Intuitively, [1] has 0 dispersion, [9/10,1/10] has some, [9/10,1/20,1/20] has more, [1/3,1/3,1/6,1/6] has more, etc.  But I haven't been able to quantify this.
I've tried using standard deviation, variance, moment of inertia, etc.  But I can't find a good way to do this.  I can arbitrarily rank order the buckets, but this seems, well, arbitrary.

(I'm the original poster, but can't seem to comment any more)
Ray: Great work, thanks for sharing this original research.  If you can provide more info on how you developed it, it would be fascinating.
whuber: Do you still feel entropy is a better measure than $k'$? Why? Intuitively, $k'$ fits like a glove.
Just noticed: The Wikipedia entry http://en.wikipedia.org/wiki/Diversity_index gives both $k'$, entropy, and other measures of diversity.  But, alas, no comparison (what are fundamental assumptions of each, when is one better suited, etc.)
 A: $k' = 1 / \sum{p^2}$ is the "effective" number of buckets over which the distribution is uniform. $1 \le k' \le k =$ the # of buckets.

Edit

*

*Thanks for adding the markup. Also, I'm new to this site, and I don't know how to reply to a comment, as opposed to replying to a post.


*I developed k' myself, sometime in the 1970's, in response to a student's request for a way of measuring the variability across categories of admission diagnoses to a mental hospital, the question being whether there was more variability at certain times than at others. Recently, I discovered that it had previously been proposed by E.H. Simpson in a short article: Measurement of Diversity, Nature, 163 (1949), 688.
k' is not restricted to uniform distributions. Calling it the effective number of categories over which the distribution is uniformly distributed is just a way of establishing the scale it's on. It's analogous to the df in repeated measures anova after correcting for non-sphericity, with the normalized eigenvalues of the covariance matrix replacing the p's.
