The Hughes Diversity Index

Ed Hughes was a recent school board member, and president, here in Madison.  He is a lawyer, and a terrific champion of public schools.  In a recent blog, Hughes describes what he calls the “Diversity Dividend”, the benefits derived by students attending schools with high racial diversity: academic, social, and civic, as well as providing better preparation for work life.

Regrettably, diversity often ends up penalizing school districts.  Parents evaluating the potential of a given school or community examine test scores as a convenient and easy measure of academic strength.  But almost invariably, students of color score lower, on average, than white students.  Consequently, the published overall average score for a given district is usually lowered as the student body becomes more diverse.

This tendency can be quite diabolical.  As example, Hughes points to the ACT scores for two local high schools, one in Madison and one in the adjacent town of Middleton.  Madison’s score for every racial group was higher than Middleton.  But because of the racial composition, the overall average for Middleton was higher; (Middleton had a higher percentage of whites, so even though they scored lower than Madison’s whites, the larger percentage helped raise their overall score above Madison).  There’s a way to report the numbers that addresses this incongruity; we’ll look at that next time.  For now, suffice it to say, you have to dig a little deeper than just one simple reported figure if you want to get a more accurate read on what the numbers have to say.

Being a blog that focuses on numbers, I wanted to describe the technique Hughes employs to measure the level of diversity.  As a starting point, he references what is called the Herfindahl-Hirschman Index, or “HHI”.  It’s an approach used in antitrust law to measure the concentration of products or services, i.e., to measure the relative lack of competitiveness.  It’s calculated by taking the market share of each participant, squaring the percentages, and adding them up.  So if a product market has four companies with shares of 50 (percent), 30, 15, and 5, the HHI would be: (50 x 50) + (30 x 30) + (15 x 15) + (5 x 5) = 2,500 + 900 + 225 + 25 = 3,650.  Another market where four firms each have a 25 share would have an HHI of 2,500 (= 625 + 625 + 625 + 625).  In its extreme form, a market with only one firm possessing a 100 share would have an HHI of 10,000 (= 100 x 100).  Clearly, the lower the score, the better: one wants more competitiveness, not less.  Similarly, to obtain the “diversity dividend”, we want to see higher diversity, not lower.  But how to measure it?

Hughes takes the HHI formula, turns it on its head, and comes up with a simple and brilliant technique for measuring diversity.  First, Hughes takes the highest possible score of 10,000, and subtracts from it the sum of the square of the racial share percentages.  Here in Wisconsin, racial mix is usually reported across five categories:  Asian, Black, Hispanic, White, & all other.  (Actually, racial mix here also includes American Indian, Pacific Isle, and Two or More Races; I’ve combined these typically very small categories into “all other”, to ensure share totals add up to 100.)  For Wisconsin, with a racial mix of 4, 9, 11, 72, & 4, that equates to:  10,000 – (16+81+121+5,184+16) = 10,000 – 5,518 = 4,482.

Next, he scales up the score so that perfect diversity adds up to 10,000.  With five categories, a perfect distribution would have all five with a 20 share, which when subtracted from 10,000 leaves 8,000.  Scaling up that score requires multiplying it by 1.25.  Stated more broadly, the adjustment entails multiplying by n/(n-1), where n is the number of categories you have.  (If you have five categories, the factor is 5/4 or 1.25; 3 categories would be 3/2 or 1.5; etc.)  The scaled up Wisconsin score is: 4,482 x 1.25 = 5,602.  To make the score less cumbersome, Hughes’ third and final step is to divide the result by 100.  So the Wisconsin score ends up simply as 56.  (Actually, the racial mixes here have been rounded; not rounded, the final score for Wisconsin comes out at 57.)

And that’s it.  I think what I’ll call here the “Hughes Diversity Index” is brilliant.  It’s easy to calculate, and so easy to comprehend and use for comparative purposes.  It can obviously be applied to any setting where you want to capture the relative mix, be it public schools, universities, corporations, communities, countries, whatever.  By the way, for the US population overall, the index is 69, while for the US student population – using averages from 4th grade and 8th grade NAEP test takers – the score is about 81.  The higher student score reflects expectations of ever-growing diversity in this country.  The Madison school district’s score was 90, the highest of all of Wisconsin’s 424 school districts.  How does your community/school district / company fare?