This is a nonsense measure of inequality

In The Conversation on Tuesday, Jennifer Chesters (University of Melbourne) wrote an interesting piece on wealth inequality in Australia. Interesting, but also partially misleading because of this bit:

Using the mean and median household wealth figures, it is possible to calculate the ratio of median to mean wealth.

The closer this ratio is to one, the lower the level of inequality. In 2003-04, the ratio was 0.63. In 2013-14, it was 0.57. This also indicates that wealth inequality increased.

The ratio of median to mean wealth doesn't tell you about inequality. It does tell you about how skewed the wealth distribution is, since the smaller the ratio is the greater the average distance from the middle of the wealth distribution the wealth of the people above the median will be. But that is not the same concept as inequality. To see why, consider these two countries:

• In Country A, every person has wealth of exactly $100,000.  The median wealth (the wealth of the person in the exact middle of the distribution) is equal to $100,000. The mean (average) wealth is also equal to $100,000. So the ratio of median to mean is equal to 1.
• In Country B, exactly half of the people have wealth of $200,000, and exactly half of the people have wealth of zero. The median wealth is again equal to $100,000. The mean wealth is again equal to $100,000. So the ratio of median to mean is also equal to 1.

It should be clear to you that in Country A, there is no wealth inequality, because everyone has the same wealth. In contrast in Country B, clearly there is wealth inequality. And yet, the ratio measure is exactly the same for Country B as for Country A. That's because neither wealth distribution is skewed - both distributions are symmetric. Ok, this was a superficial example, but it should be enough to illustrate that the ratio of the median to mean wealth is nonsensical as a measure of inequality, since skewness and inequality are not the same thing.

Fortunately, the ratio of median to mean isn't the only measure that Chesters uses:

The P90 to P10 ratio compares the wealth of households at the 90th percentile with that of households at the tenth percentile. A larger ratio indicates greater levels of inequality.

In 2003-04, households at the 90th percentile held 45 times as much wealth as households at the tenth percentile. In 2013-14, households at the 90th percentile of the distribution held 52 times as much wealth as households at the tenth percentile. This indicates that wealth inequality increased in that decade.

These sorts of ratio methods are crude, but simple to calculate (which is why we use them in my ECON110 class). They also have problems, chiefly that they ignore most of the people in the distribution - for example, the P90 to P10 ratio is the same regardless of whether the top person in the distribution has wealth of $100 million, or $100 billion. However, at least they aren't the nonsense that the ratio of median to mean wealth is.