Quantifying Inequality

Occupy Math has heard many, many claims about wealth, who has it, who is being deprived of it, and the problems that unequal distribution of wealth causes.  In this edition of Occupy Math we look at a tool for quantifying the degree of income inequality in a country or area, the Gini Index.  This mathematical tool was invented by Italian sociologist Corrado Gini.  The world bank, the CIA, and a number of other organizations use the index to quantify the way wealth is distributed in a country.

Why is Occupy Math interested in this measuring tool?  Horrible events from the French Revolution to the 2016 American Presidential Election have involved inequality of wealth in a fundamental way.  French aristocrats died horrible, bloody deaths — driven by the incredible misery they inflicted on their subjects.  The decision of a substantial fraction of the American electorate to support an obviously unqualified, over-privileged candidate whose speeches and claims are detached from reality is supported, ironically, by the fact that wealth in the United States has become highly concentrated — making it harder for the government to serve the general welfare.

The Gini Index is a simple, effective single-number measurement of the irregularity of distribution of wealth.

As Occupy Math noted in the post People that REFUSE to do the Math: math-free public policy isn’t just ineffective, it is often counter-productive.  The ability to measure and track the distribution of wealth is an important public policy tool.  People with math skills can protect and preserve their freedom more effectively.


The Gini index is based on the way wealth is distributed.  The horizontal axis of the graph is the fraction of the population while the vertical is the fraction of the wealth.  The yellow part of the diagram shows how the wealth accumulates as the fraction of the population grows.  If everyone had the same amount of wealth then the yellow part of the graph (“B“) would be a triangle filling the lower right part of the square.  If one person had all the wealth then no yellow would appear — the “A” portion would be the entire graph.

The value of the Gini index is twice the area of the blue part of the plot.

If wealth in a society is evenly divided then there is no blue and the Gini index is 0%.  If one person has all the wealth then the lower triangle is all blue — with an area of 1/2 — then the Gini index is 100%.  The more unequal the income distribution, the closer to 100% the Gini index grows.  Here are three example distributions — notice how the blue portion of the graph grows with the inequality.


The curve in these diagrams that shows how the wealth is distributed is called the Lorenz curve.  There are, in fact, a number of possible Lorenz curves you could use depending on what type of wealth you are measuring.


Let’s compare some countries using estimates of the Gini index by the world bank.  The lowest is Ukraine with 24.6%; the highest is Comoros with 64.3% — a very poor country with a few very wealthy people.  This gives us a sense of the practical range of the index.  Let’s look at a few more countries.

Sweden: 27.3%
South Korea: 31.3%
United Kingdom: 32.6%
Canada: 33.7%
United States: 41.4%
Russia: 41.6%
Malaysia: 46.3%
Mexico: 48.1%

Careful readers will notice that several of these figures are out-of-date; with that caveat, think about what you know about these countries and see if there are any patterns in the Gini index.  There is also a list of the Gini index for American states.  The best state for even distribution of income is Utah while the District of Columbia hits a grossly uneven 53.2%.  Washington DC mixes great wealth and great poverty to a larger degree than any other place in America.

You cannot fix a problem until you can measure progress toward a solution.

On the date of the release of this post, many of Occupy Math’s readers will have seen an incredible number of false, recent claims in the media — partly in connection with the election, but there is a very high rate of background inaccuracy in the media.  Math is an important tool for sorting out truth from fiction.  There are many topics that math cannot help with, but there are many topics where it does help.  Suppose, for example, that a major city starts an anti-poverty program to help raise the rate of employment and average income of its citizens.  Two years later, the program needs to either be funded again or shut down.  By the time this decision is made, there will be many people with a vested interest in the program and other that hate it.  How to you make a good decision?  In an American city, the Gini index is fairly far above zero.  A decrease in this index would indicate that the anti-poverty program is working (or that the rich had gotten poorer, a somewhat less likely outcome).

While the Gini index was developed to measure the degree to which wealth is distributed unevenly, it has other uses.  Machine learning is the discipline of automatically pulling patterns from data.  A standard technique is to build a decision tree that breaks the data into groups of similar cases or situations.  The Gini index can be used to decide which variables divide up the data, or a subset of the data, most evenly and so create the most effective decision tree.

This post has taken a trip to the land of quantitative social justice — a largely unfamiliar place that we would benefit from exploring to a greater degree.  While the Gini index was created to measure inequality in income, it can measure inequality in anything and so is a general purpose mathematical tool.  Do you have examples of quantitative tools you would like to see a post on?  Feel free to comment or tweet.

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics


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