Below average

Which measure of income best gauges a nation’s economic success?

To characterize a dataset simply, economists usually figure the average value (the “mean”).  If the Budapest String Quartet dined at the Four Seasons individually for $50, $75, $100 and $25 (that one for a glass of water), then one may measure their average gastronomical pleasure as the mean of their checks -- $250 divided by 4, or $62.50.  In general, the mean for n data points xi in a set is the sum of x1, x2, x3, …, xn, divided by n.

The mean is particularly useful when the likelihood of some number depends on its distance from the mean, regardless of whether it is higher or lower.  For example, if the mean grade in a classroom is a C, then the likelihood of an A, which is two letters higher than the mean, may equal the likelihood of an F, two letters lower than the mean.  In the presence of such symmetry, the mean can tell us a lot.  (And we could learn a lot more if we had a measure of how widely dispersed the grades were.  For example, are many grades Bs and Ds, or do they tend to be As and F’s?  But for now, let’s ignore dispersion).

Not all data follow a symmetric distribution.  In most nations, household income is skewed to the right, because we have a few billionaires.  Their treasures push up the mean income to a point that most of us consider unrealistic for a “typical” family.  It would seem to make more sense to use, as a measure of average income, the figure that is just above 50% of the household incomes (and, of course, just below the other 50%).  This is the median.

A trifling' trillion

The median has other fortes.  In a dynamic economy, the distribution of household incomes may change every few years.  It will remain skewed to the right, but the length of the skew may vary.  (Just wait until we pick up a few trillionaires.)  If we want a summary statistic that will hold true for more than a few years, then the mean won’t do.  The trillionaires will cause it to skyrocket.  The median, however, won’t change as long as it is just below half of the household incomes, regardless of how large some of those incomes may be.

Consequently, a growing gap between mean and median income may tip us off to rising inequality in the distribution of income.  In the United States, mean family income grew much more rapidly than its median counterpart over the Eighties and Nineties, concluded Arnold Katz of the Bureau of Economic Analysis in the U.S. government.  Over the long run – from 1969 to 2009 – mean family income grew 1.08% per year; median family income, only .69%.  Both figures were adjusted for price changes.

In Kazakhstan, mean income grew far more rapidly than in Europe and Central Asia in general for several years up through 2012, when it was $9,780 for KZ, reported the World Bank (the Notes have details).  This raises the possibility that income inequality grew relatively rapidly in Kazakhstan.  Without more data, it is only a possibility.

What does the mean mean?

The median has a hidden weakness:  It cannot reflect uncertainty about the income of a given household.  As we know all too well, the amount of money we make depends partly on chance.  The owner of a Cajun restaurant in New Orleans early in 2005 could not have predicted that a hurricane would blow away her assets.  Surely our notion of average income should take such uncertainty into account.  We may want a summary statistic for the income of each household – and then a statistic that summarizes all of those statistics. 

In that case, the median would not suit us, because it does not reflect that every household income may vary according to circumstance.  It just gives us the income that is at the midpoint of the entire distribution.  Yes, we could compute a median for each household, since its income varies from one scenario to another; but it is not clear how all of these medians would relate to the national one.  For the mean, however, the relationship is crystal clear.  The mean of the distribution is the sum of the means of all the households; it takes individual uncertainty explicitly into account.  Suppose that mean income is $10 for one person and $30 for another.  Then the mean income for this distribution of two people is $40 divided by 2, or $20.

So is mean income a better measure of economic success than median income?  Not necessarily.  Consider an economy of three households.  Family A earns $10,000; B, $20,000; and C, $3 billion.  For the distribution, the median is $20,000, and the mean is just over $1 billion.  Does the high mean mislead us about the economy’s success?  Yes, if C had opportunities that A and B didn’t.  The median would be more accurate.  No, if all three families enjoyed the same opportunities but fortune smiled on C.

The point, of course, is that we should provide both the mean and the median of the income distribution – and let the user choose.  The charge that conservative analysts use the mean in order to overstate the economy’s success is not always justified.  Neither is the corresponding charge against liberals.  –Leon Taylor, tayloralmaty@gmail.com

Notes

1.        The World Bank adjusted its estimate of mean income in Kazakhstan, which is in U.S. dollars, by using a three-year average for the exchange rate and by controlling for differences in inflation rates between Kazakhstan and a composite of the euro area, Japan, the United Kingdom, and the United States.  This is the Atlas method.  It tries to avoid sharp fluctuations in the exchange rate that occur in the short run for reasons unrelated to economic capacity.
2.  To measure the inequality of a national distribution of income, economists widely use the Gini coefficient.  It ranges from 0 to 100, where higher numbers denote more inequality.  (In principle, the coefficient ranges from 0 to 1, but practitioners often multiply this by 100 for easy comparisons.)  The coefficient measures the disparity between a group's share of the population and its share of income.  If x percent of the population always has x percent of national income, then the coefficient is 0, denoting perfect equality.  On the other hand, if 99% of the population has 0% of the nation's income, then the coefficient is nearly 1.

The World Bank reports a coefficient of 29 for Kazakhstan in 2009.  By comparison, the coefficient was 40.1 for Russia that year -- and 54.7 for Brazil, a country famous for income inequality.  Gini coefficients elsewhere in Central Asia are fairly low:  36.2 in Kyrgyzstan and falling; 30.8 in Tajikistan.

Good reading

Bulmer, M.G.  Principles of statistics.  Dover.  1979.  A genial introduction.

Katz, Arnold J.  Explaining long-term differences between Census and BEA measures of household income.  Bureau of Economic Analysis.  2012.  bea.gov 

World Bank.  World Development Indicators.  worldbank.org