The Gini Coefficient — But Don’t Run!

This post originally appeared, in 2008, on an earlier version of LaMarotte. It is reprinted here without changes.

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The Gini Coefficient measures income inequality in a country—or any region—using a single number. It was created by Corrado Gini, an Italian statistician, in 1912. I’ve know of it for a long time—and it has irritated me at right regular intervals. Why? I was taught by one of my wise elders, then a junior in business analysis, that any number that people produce in such analysis should present all of the data necessary to replicate it with a simple calculator. My guru, who then labored as the chief of statistics for Anheuser Busch, carried his calculator strapped to his suit-belt. He practiced what he preached. But when we look for an explanation, we’re bowled over by references to the Lorenz curve and presented with stuff that looks like this:

G = 1 – 2 ∫_o¹ L(X)dX

The last time this happened to me (yesterday) my irritation  produced a determined search for a simple explanation. Therefore I am now prepared to explain the Gini Coefficient in plain language, namely how it is actually calculated and how the data, to be used, must be arrayed for the calculation.  I’ll use the following chart using U.S. household data for 2008 for this explanation.

The raw data for this chart, which I’ve taken from this Census Bureau site, shows the cumulative share of household income as we proceed from the poorest toward all households, thus from the lowest fifth of all households up the line until all households are included.  That is the blue line. Here is the way we must read the chart. The lowest 20 percent of households (lowest quintile), accounts for 3.5 percent of total income. The lowest 40 percent of households (the lowest and second quintile cumulated) account for 12.1 percent of total income… And so on to the last column where—surprise—all households account for all of the income. Clear so far?

The blue line represents actual results for 2008. The red line, by comparison, shows what the results would be if every group earned exactly the same amount. Not surprisingly, 60 percent of all households, then, would be earning 60 percent of the total income. This is not rocket science either.

Notice now the area surrounded by these two curves. It represents the inequality in income, thus the difference between an ideal and an actual state of affairs. What the Gini coefficient (also called an Index or a Ratio) actually calculates and reduces to a single number is the magnitude of this difference.  I’ll present the formula and how its elements are obtained. I have not penetrated deeply enough to explain the formula itself.

To begin with, we make note of the last number—100 percent in our case. We’ll call that T for Total. Next, we calculate a value called Sigma. It consists of the sum of all of the numbers added together—up to but excluding the last. In our case that is 3.5 + 12.1 + 26.7 + 50.9 = 93.2. That is Sigma. Finally we note the number of groups we used in the analysis. We used quintiles, therefore we used five groups. We generalize that number by calling it n. Now we insert these values into the formula used to obtain the Gini Ratio. That formula is:

Gini = 1 – (2 divided by T times Sigma + 1) divided by n.

Translated into numbers, this means Gini = 1 – (2/100 * 93.2 +1) / 5.  The result of this calculation is 0.4272. That’s the Gini Ratio. You may encounter it multiplied by 100 for easier readability (here 42.7).

If we apply the same approach to the top line, we have a T=100, Sigma = 200 and the formula becomes Gini = 1 – (2/100) * 200 + 1) / 5. This results in 0. In the ideal case, in other words, there is zero inequality.

Having followed this procedure, we have now generated a single number for each curve and we can therefore compare them. The rule here is:  The lower the Gini the more equal is the income distribution. It can’t get any lower than zero–and can never exceed 1. A result of 1 would mean that a single group has all the income and nobody else has any.

Let me follow this up by looking at the Gini Ratio over some period of time. The following graph (its source is here) does that for us for the period 1967 to 2007.

Income inequality, although it rises and falls year to year, has been increasing steadily over the recent forty year history presented above. The Gini is useful especially at this level of macro analysis. It holds a vast amount of detail in a single number. And now that I know how it is obtained, I find it much more acceptable. [For an updated Gini Ratio to 2013, see the previous post here, same date.]

An additional note. Country to country comparisons using Gini calculation are interesting but not much more than that. Several organizations (the CIA and UN are two) calculate this number for many countries. The U.S. falls generally into the upper ranges of inequality–but not at the very top. To find the peaks, we can single out Brazil and Mexico. China? China’s inequality is just about the same as ours. And Japan’s falls below ours. Bulgaria is hugging the bottom range–at least in the list of countries shown in this Wikipedia chart.

Income and Inequality: Update to 2013

Back in 2011 I presented here graphics on Household Income (Average and Median) and on Income Inequality (as measured by the Gini Index). Herewith an update of those graphics.

The data in this chart are in 2013 constant dollars and obtained from this BLS facility (Table H-6, All Races). Median here means that half of all households earn less, half earn more—therefore it is the income at the precise middle of the total earnings range. Note how close average and median are to one another in 1975 ($7,800) and how that difference has grown by 2013 ($20,702)—and this in constant dollars, thus in actual purchasing power. The difference is accounted for by growing income inequality, which brings us to the next chart.

In this graphic I reproduce the Gini Coefficient which measures inequality. A Gini Index of 0.0 means absolute equality of all incomes. Therefore the higher the value the greater the inequality. The data for this graphic are from this BLS facility; select Income Inequality and then Table H-4. Posts on how the Gini is calculated will be put up here in due time; they were on the old LaMarotte which is no longer accessible.

Worth noting here is that the Great Recession affected Inequality only briefly, by causing it to drop somewhat between 2007 and 2008—significant numbers of the very rich lost income in the housing crash. The Gini has grown since although, in the 2011 to 2013 period, it has shown signs of lessening. I’ll revisit this picture a year or so later when new data are published. For the time being, the rich are getting richer…. So what else is new?

Employment Update: January 2015

At this time of the year, labor force numbers are presented, by the Bureau of Labor Statistics, in fully revised form—with the revisions reaching years back in time. This year’s revision do not change the overall pattern of developments except in one regard. Numbers for 2014 have been revised upward significantly. In my last report I showed job gains for 2014 at 2.96 million; the revised numbers lift the year to 3.12 million. Since 2014 was better than initially reported, total job losses of 8.7 million (not changed) were recovered earlier (by December 2013) than earlier reported (by May 2014). It took us four years to recover all of the jobs lost in 2008 and 2009.

Job gains in January 2015 were 257,000, below the (revised) 329,000 (versus last month’s report of 252,000). Therefore the projected annual results for 2015 are presently lower than the actual 2014 results, but it’s early days yet. The January figure this year is better than in all earlier years, except 2012, since the Great Recession.

Herewith the two charts I usually show—month by month and then annualized. In the second chart, the 2015 figure is a projection.

Since the economy has, since the end of 2013, almost entirely caught up with job gains needed to support population growth as well (about 80 percent), the motivation for this series is beginning to fade. We are recovering. But it took a long time.