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  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,  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.)