Chapter 6.4, Problem 120AE

Diversity Index Shannon’s diversity index is a measure of the diversity of a population. The diversity index is given by the formula

$H\text{ = }-\left(p_1\log p_1\text{ + }{p}_2\log p_2\text{ + }\mathrm{...}\text{ + }{p}_n\log p_n\right)$

where $p_1$ is the proportion of the population that is species 1, $p_2$ is the proportion of the population that is species 2, and so on. In this problem, the population is people in the United States and the species is race.

(a) According to the U.S. Census Bureau, the distribution of race in the United States in 2010 was as follows:

Compute the diversity index of the United States in 2010.

(b) The largest value of the diversity index is given by $H_{\max }\text{ = log}\left(S\right)$, where $S$ is the number of categories of race. Compute $H_{\max }$.

(c) The evenness ratio is given by $E_H\text{ = }\frac{H}{H_{\max }}$, where $0\le {\text{ E}}_H\le \text{ 1}$. If $E_H\text{ = 1}$, there is complete evenness. Compute the evenness ratio for the United States.

(d) Obtain the distribution of race for the United States in 2000 from the Census Bureau. Compute Shannon’s diversity index. Is the United States becoming more diverse? Why?