An article in the Admin. Sciences Quarterly, Vol. 54, June 2009 at 282, describes an index to measure how even or skewed distributions of nonprofits funding were from multiple sources. The same formula should describe the distribution of spending by a legal department on, say, the 20 law firms on which it spends the most during a fiscal year. [I picked 20 simply to create a cutoff point and because most law departments in fact engage 20 or more law firms each year.] If several departments computed the figure, we could have a benchmark figure for concentration of outlay on law firms.
The formula is
where n denotes the total number of law firms paid (20) and Si is the proportion of spending on the ith law firm in the total spend (for example the percentage of total fees paid to the 20th firm). [This blog can’t show the little i=2 under the brackets or n over the brackets]
Let’s assume for a particular law department that the 20th firm down the list received 0.5% of the total spend of firms on the list. Thus, (1/30-0.5) squared *30/29= 0.00115. You do that calculation for each of your top 20 firms and subtract the total sum from one. The Greek letter sigma means you subtract from one the sum of all those calculations for each of the 20 firms.
A law department that spends evenly on all its 20 primary firms would score 1. With this formula, law departments would have comparable figures for benchmarking dispersion of spending on outside law firms.