Diversity Index

[mark...@twc.com]

https://newyork.hometownlocator.com/counties/subdivisions/data,n,town%20of%20palmyra,id,3611756198,cfips,117.cfm 

"The Diversity Index is a scale of 0 to 100 that represents the likelihood that two persons, chosen at random from the same area, belong to different races or ethnic groups. If an area's entire population belongs to one race AND one ethnic group, then the area has zero diversity. An area's diversity index increases to 100 when the population is evenly divided into two or more race/ethnic groups."

This definition of "diversity index" seems (at first blush) okay, but might a different definition be better?

Let's say a town is exactly half White Mormons and exactly half Black Baptists.  Then its diversity index is (1/2)*(1/2)+(1/2)*(1/2) = 1/2 = 50 (%).  This disproves the last sentence, "An area's diversity index increases to 100 when the population is evenly divided into two or more race/ethnic groups."

If the town consists of n ethnic groups, each accounting for 1/n of the total population, the town's diversity index is (1/n)*((n-1)/n) = (n-1)/n^2.  When n=10, the diversity index is 9/100 = 9 %.

Wait!  The town is highly diverse, but its diversity index approaches 0 it becomes more diverse (as n approaches infinity).  

This is a  very bad definition for a measure of diversity!