# Ask Gini: How to Measure Inequality

## Frank Cowell, an economist at the London School of Economics and Political Science, says the Gini coefficient is like the Kardashians: "It's famous for being famous."

https://www.scientificamerican.com/article/ask-gini/#

Ask Gini: How to Measure Inequality

By Evelyn Lamb on November 12, 2012

Credit: Wikimedia commons/Reidpath (public domain)

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Frank Cowell, an economist at the London School of Economics and Political Science, says the Gini coefficient is like the Kardashians: “It’s famous for being famous.” He’s speaking about one of the most commonly discussed measures of income and wealth inequality. The Gini coefficient has been in the news a lot since the U.S. Census Bureau released its most recent data on income inequality in September. The data show that income inequality in the U.S. is high, but many articles blunder when they try to compare the U.S. to other countries.

An article on the Atlantic Web site in October, for example, reports on a pairing of U.S. cities with foreign countries that have similar Gini coefficients. The city in the U.S. with the least income inequality, Ogden, Utah, was paired with Malawi in Africa, whereas the city with the greatest inequality, Bridgeport, Conn., was paired with Thailand in Southeast Asia. These pairings are a bit puzzling. Is it better to be Malawi than Thailand? Does it make sense to compare the Gini coefficient of one concentrated metropolitan area with that of an entire nation?

A closer look at the data used to create the map shows that, as reported in a Forbes editorial, the U.S. Census Bureau usually reports Gini coefficients based on pretax numbers, whereas many calculations for foreign countries use posttax numbers, which often include redistribution of wealth from rich to poor and tend to lower the Gini coefficient. Comparing the pretax number in one country with the posttax number in another is somewhat meaningless.

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To understand what the Gini coefficient can and cannot explain, and how to interpret articles about economic inequality, a deeper look at this statistic is required.

The Gini coefficient compares the income or wealth distribution of a population to a perfectly equal distribution—in which every citizen of a city or country has equal wealth. To compute the Gini coefficient, economists first find the Lorenz curve for the population. The curve is a graphical representation of the distribution of income or wealth in a society. The x-axis is the proportion of the population, from lowest to highest income, and the y-axis is the cumulative percentage of income or wealth owned. So the point (0.5, 0.2) would indicate that the lowest-income 50 percent of the population earned 20 percent of the total income. A perfectly equal society would have a Lorenz distribution that looks like the line y = x.

The Gini coefficient measures how far the actual Lorenz curve for a society’s income or wealth is from the line of equality. Both the Lorenz curve and the line of equality are plotted on a graph. Then the area between the two graphs is computed. The Gini coefficient is the area between the two graphs divided by the total area under the line of inequality. In the picture at the top right of this article, it is the area of the region labeled A divided by the combined areas for A and B. This yields a number between 0 and 1, sometimes reported as a percentage—for example, 0.22 or 22 (written with or without the percent sign). 0 means that the country is perfectly equal, and 1 means that one person has all the wealth or income. (This Web site has a Lorenz curve generator and Gini coefficient calculator: Enter a set of incomes to find out what the Gini coefficient of the group is and what the distribution looks like.)

Cowell says that the Gini coefficient is useful, particularly because it allows negative values for income and wealth, unlike some other measures of inequality. (If some amount of the population has negative wealth (owes money), the Lorenz curve will dip below the x-axis.) But the Gini coefficient also has limitations. For one, it takes all the data from the Lorenz curve and converts it to a single number. Two different income distributions can have the same Gini coefficient, and a lot of information is lost in the conversion to a graph. Cowell asks, “Why not just look at the Lorenz curve?”

In addition, the Gini coefficient cannot tell that person X is a 24-year-old medical student who has negative income because of student loans, whereas person Y, who has the same amount of negative income, is unemployed and without job prospects. It samples people at random points of their lives, which means that it can’t separate those whose financial futures are reasonably secure from those who do not have prospects. Its results are also sensitive to outliers—a few very wealthy or very poor individuals can change the statistic significantly, even in a large sample.

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Cowell says that the Gini coefficient should not be used as the sole measure of economic inequality. He suggests two ways to handle the number: “One is to look beyond the Gini as a single statistic. The other is to consider whether it might be useful to use a model of the upper tail of the distribution, so you get a clearer picture.” Due to incomplete data, the Gini coefficient can underestimate the concentration of wealth in the very richest individuals, and can even underestimate the wealth inequality within the upper echelons of the wealthy. To mitigate this problem, Cowell studies better ways to model income and wealth distribution in the most well-off. One option is to “patch in” an assumed distribution (specifically a Pareto distribution) for the top 5 or 10 percent of the population. In effect, this means assuming that the distribution of wealth takes a certain form, and using that model, rather than sparse data, to calculate the Gini coefficient.

The study of income and wealth inequality are of course fertile ground for many questions and controversies. What “should” the inequality in a society be? As the Occupy Wall Street (OWS) movement highlighted, this is of broad interest. Cowell says that the concentration of wealth in the upper echelon of the population can be reminiscent of a monopoly in business. “People get kind of twitchy if a large portion of the output is controlled by a small number of firms.” Economists are asking many different questions about the causes and effects of wealth and income inequality. What are the effects of high inequality, and is it possible to separate the effects of poverty from the effects of inequality?

But the Gini coefficient is not just used by economists. Sam Shah, a high school math teacher in Brooklyn, N.Y., wrote in his blog that he included a section on the Gini coefficient during the last week of his calculus class. (He based his lesson on this handout (pdf) from the North Carolina School of Science and Mathematics.) He framed the lessons in the context of OWS and asked, “Is income truly becoming more and more unequally distributed in the past 40 years?” Students had access to several decades’ worth of data and got to explore questions about how they thought income and wealth should be distributed, in addition to working on math. As a way to motivate students’ desire to understand calculus and statistics, he found the Gini coefficient to be very effective.

“The best part of the discussion was around what kids picked for ‘what they would like it to be,'” wrote Shah. Others have studied this question: How do people feel about wealth inequality? A 2005 survey (pdf) conducted by Michael Norton of the Harvard Business School and Dan Ariely of the Duke University Department of Psychology and published in 2011 found that most Americans underestimated the amount of wealth inequality in the U.S. and wanted it to be even lower than their estimates. The researchers showed respondents three different pie charts illustrating possible wealth distribution by quintiles. One illustrated complete equality, one was slightly unequal—with the lowest quintile earning 11 percent of the wealth and the highest earning 36 percent—and one was based on the wealth distribution of the U.S., with the lowest quintile owning 0.1 percent of the wealth and the top quintile owning 84 percent. Of the people surveyed, 47 percent preferred the slightly unequal distribution, 43 percent the perfectly equal distribution and only 10 percent the highly unequal distribution.

The diagram in the article labeled the slightly unequal distribution as Sweden, although it was presented without a label to survey respondents. The authors clearly wanted readers to believe that Sweden’s wealth distribution was preferable to that of the U.S.—a heading in the article stated “Americans Prefer Sweden.” But in a note at the end of the article, the authors wrote, “We used Sweden’s income rather than wealth distribution because it provided a clearer contrast to the other two wealth distribution examples; although more equal than the United States’s wealth distribution, Sweden’s wealth distribution is still extremely top-heavy.” (According to Cowell’s research, even that statement is unclear: some methods of computing the Gini coefficient that include a modification of the distribution at the top of the wealth scale find that Sweden has greater wealth inequality than the U.S.)

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Although Norton and Ariely’s conclusion that Americans would prefer a more equal distribution of wealth may be sound, their data switcheroo illustrates a common problem when talking about inequality: income versus wealth. People often conflate the two, but they are not the same. “You’re using your thermometer to measure something quite different,” Cowell says. Wealth inequality says more about the balance of power in a society, and income inequality addresses the way labor markets operate. “Typically wealth is much more unequally distributed than income, in any country you look at,” he says. He recently conducted a study (pdf) about income and wealth inequality in the U.S., U.K., Canada and Sweden. Because Sweden has more interventionist policies, one might assume that the U.S. would be much more unequal than Sweden. “It’s true for income but not for wealth,” says Cowell.

Economists continue to use the Gini coefficient, either standard or modified, to understand wealth and income inequality. In the meantime, lay people who want to understand wealth inequality should read the fine print to ensure that they have all the facts.