# Inequality: Statistics Question, from Kim Zussman

September 28, 2010 |

The distribution of income is bounded on the low end by zero, but unbounded on the high end. This resembles the distribution of stock returns, and is better described by log-normal distribution.

Presumably humans evolved to anticipate something like normal distributions bounded by zero and bounded on the high end; height or weight for example. If heights were distributed like income, most of the time you would encounter normal-looking people, but occasional 20 footers. Of course tribes of average folk would to try hard to befriend the big guys.

## Phil McDonnell writes:

I think there is a statistical quirk. Namely the quintiles are reconstructed every year with new individual members. Thus the 2009 top quintile contains different people than the 2010 top quintile. To understand how this creates a bias we need to look at how new people enter and leave the top quintile and what that process does to the 'average' of the quintile. To enter the top quintile the individual can only come from below. Thus he lowers the average of the quintile below and possibly raises the one above. No one can leave the top quintile by rising out of it. Thus there is an upward bias in the sense that they are retained no matter how high they go. On the other hand they are eliminated if they fall in income.

In the bottom quintile the reverse is true You cannot leave by going to zero, you are still in the bottom quintile. But if you make too much you will move up to the next quintile and thus reduce the average in the bottom quintile.

The middle three quintiles have less bias in this sense because individuals who leave can either go up or down to the next quintile resulting in more of a wash. In the same fashion new entrants to a given quintile can come either from above or below again resulting in more a a net wash effect.

The comparison of the top quintile to the bottom inevitably results in a biased and distorted comparison because of this effect. It would be better if they compared the second from the top and second from the bottom quintiles to reduce the bias. Reducing the bias is probably not the goal of those who calculate such statistics.

This is a different question that relates to what the statistics represent and will be used for. What Phil writes is certainly correct. What the quintiles show is the income distribution at any one point in time. It does not tell you anything about lifetime income or the ability to better one's self over time, that is upward mobility in the quintiles, or the fact that some of the well off become less so over time. For that you would need other measures. The movements between the groups will create the biases described. But to say that the bottom fifth only earn so much over time x and the top fifth earn so much need not have an upward bias to what these statistics actually measure as such movement happens all the time to varying degrees over time and by country. The top fifth are still the best off and the bottom fifth the worst off. Were they stand an any one time is what it is and that is all this statistical approach shows. There is no need to correct this bias but one does have to develop other measures to answer the sort of questions that seem to concern those who point to bias. There is no statistical reason why the growth rates have to favor the top group. That tendency to the extent it exists is due to political economic factors, cultural factors, social factors, etc.