By Tim Hickey, posted June 6, 2016 —

Students in a calculus class are not expecting to engage in a discussion on the causes of the 1929 stock market crash. So I like to begin my class on applying the area bound by two curves by facilitating that very discussion. Students typically can describe many factors (a credit to my history-teaching colleagues!), such as overspeculation in the stock market, an overreliance on credit, a languishing agricultural economy, stagnant wages, or inflated tariffs. We then read a passage from the excellent “nutshell” book by John J. Newman and John M. Schmalbach titled United States History: Preparing for the Advanced Placement Examination (1998). The passage proposes “uneven distribution of income” as one of the causes of the stock market crash. It continues: “Economic success was not shared by all, as the top 5 percent of the richest Americans received over 33 percent of all income.”

But what about the top 25 percent or the bottom 10 percent? How does this distribution compare to other societies or time periods? The single statistic is lacking. Is there a way to describe the overall equitability of the distribution of some resource in a society in a concise way? The answer, it turns out, is yes. The number that such organizations as the United Nations and the Central Intelligence Agency use is called the Gini Index.

Calculus, and specifically, the area bound by two curves, is used to generate the Gini Index. In the lesson, I posit a hypothetical neighborhood with 5 families and then use it to generate the cumulative percentage of total income as a function of the cumulative percentage of population. We then graph the resulting “Lorenz Curve” for the neighborhood and use a power regression to find the equation. Next, we graph the line y = x, which represents a society with a perfectly equitable distribution. The Gini Index (at least a simplified version of it) is the ratio of two areas—the area bound by a Lorenz Curve and the line y = x compared to the area bound by the line y = x and the x-axis (this latter area is, of course, just 1/2). This ratio generates a number that is between 0 and 1, where 0 would represent perfect equity and 1 would represent perfect inequity. For the hypothetical neighborhood, the Gini Index is 0.2942. The number is built on data representative of the whole neighborhood, and so it is better than the isolated statistic presented in the nutshell history book. But comparison of data makes the index more meaningful. Thus, I present a list of countries (and the United States at various points in time) with raw data collected by the United Nations in 2008, and allow students to work in groups to calculate the Gini Index for countries of their choice. We then typically compare our data and have a rich, joyful, and I hope, inspiring discussion about the usefulness of calculus!

Tim Hickey is a Nationally Board Certified Teacher and the math department chair at Monticello High School in Charlottesville, Virginia.