I don't know much about statistics, besides a couple of humorous chestnuts, so please bear with me.
Let's say I run a corporate tax law firm in Georgia with two other white guys. We have a hundred plus associates, of which four are black, four are Asian and there's a Hispanic guy. Not the best showing of diversity, but it's much better than that of our nearest competitor which has one associate who looks vaguely Asian. I would like to say my firm has a higher "diversity score" than my competitor.
This diversity score should measure how much or how little the racial breakdown of the lawyers in my firm deviates from the racial breakdown of all the people in Georgia. The first way I thought was simply to take the percentage of a given race in Georgia and subtract the percentage in my firm, then add it all up. This would give negative points for the over-representation of whites (my firm's 91% compared to Georgia's 63%) but it seems inappropriate that it would give positive points for the under-representation of blacks (31% in Georgia compared to 4% in my firm). It's possible that with this half-baked method certain over-representations could cancel out the effect of under-representations.
There has to be a better way to compute this diversity score, whether "off-the-shelf" or invented for this purpose.
This is probably way more in-depth than you're interested in, but John Baez has blogged about different measures of biodiversity. There are at least $8$ parts, and you can read the first of them here. It is a whole interesting saga!