joel
November 21st 2011, 05:57 PM
So I've seen graphs that show that the richest x% of people make more than x% of the income. Like so:
70711
And hockey-stick shaped graphs (http://www.lcurve.org/) that are supposed to show how ridiculously more some people earn than others. See also http://visualizingeconomics.com/2007/02/15/2005-us-income-distribution-part-3/
But I was curious how much this is just a fluke of statistics.
For example, consider human height. It is just a mathematical necessity that the tallest x% have more than x% of all the height. If you have 100 random people, and you add together the heights of the tallest 10, the sum must be greater (and probably a good deal greater) than the sum of the heights of the shortest 10 people. And height, as I recall, falls into a standard bell curve.
So I thought I'd consider what would happen if incomes fell in a bell curve.
Suppose the peak of the bell curve is at 0 and falls off to the right (with increasing income on the x axis). It is necessarily the case that the:
Top fifth: make about 44% of the income
Next fifth: 26%
Middle fifth: 17%
Next fifth: 10%
Bottom fifth: 3%
The numbers are the same no matter what standard deviation you choose, because the standard deviation in this case is entirely determined by your choice of units: dollars, yen, gold oz, etc.
Thus these numbers are the mathematically necessary property of the normal distribution. Thus one shouldn't be surprised at all if the real-world percentages have a similarity.
There is also no sense in complaining that there is some person making many times what even the 98th-percentile guys do, thus forming a hockey-stick (L-shaped) curve.
By drawing this curve for the bell curve, as you take smaller and smaller intervals the graph approaches the L-curve. In fact it's even worse, because the mathematical bell curve stays positive all the way to infinity (income). Thus no matter what point on the bell curve you pick, you can always find some percentage of people that make a quadrillion times that much, thus you can always make an L-curve from it as sharp and extreme as you could imagine.
So these are just the statistical artifacts one should expect from a normal distribution.
(This all having been said, I might add that the actual income distribution will likely be different from the normal distribution as you approach 0. As seen here: http://visualizingeconomics.com/2006/11/05/2005-us-income-distribution/ Some other mathematical function (perhaps the Gamma distribution http://en.wikipedia.org/wiki/Gamma_distribution) would be a better fit. But the qualitative things I've discussed here would still apply, since you have a distribution that asymptotes to zero as income goes to infinity.)
70711
And hockey-stick shaped graphs (http://www.lcurve.org/) that are supposed to show how ridiculously more some people earn than others. See also http://visualizingeconomics.com/2007/02/15/2005-us-income-distribution-part-3/
But I was curious how much this is just a fluke of statistics.
For example, consider human height. It is just a mathematical necessity that the tallest x% have more than x% of all the height. If you have 100 random people, and you add together the heights of the tallest 10, the sum must be greater (and probably a good deal greater) than the sum of the heights of the shortest 10 people. And height, as I recall, falls into a standard bell curve.
So I thought I'd consider what would happen if incomes fell in a bell curve.
Suppose the peak of the bell curve is at 0 and falls off to the right (with increasing income on the x axis). It is necessarily the case that the:
Top fifth: make about 44% of the income
Next fifth: 26%
Middle fifth: 17%
Next fifth: 10%
Bottom fifth: 3%
The numbers are the same no matter what standard deviation you choose, because the standard deviation in this case is entirely determined by your choice of units: dollars, yen, gold oz, etc.
Thus these numbers are the mathematically necessary property of the normal distribution. Thus one shouldn't be surprised at all if the real-world percentages have a similarity.
There is also no sense in complaining that there is some person making many times what even the 98th-percentile guys do, thus forming a hockey-stick (L-shaped) curve.
By drawing this curve for the bell curve, as you take smaller and smaller intervals the graph approaches the L-curve. In fact it's even worse, because the mathematical bell curve stays positive all the way to infinity (income). Thus no matter what point on the bell curve you pick, you can always find some percentage of people that make a quadrillion times that much, thus you can always make an L-curve from it as sharp and extreme as you could imagine.
So these are just the statistical artifacts one should expect from a normal distribution.
(This all having been said, I might add that the actual income distribution will likely be different from the normal distribution as you approach 0. As seen here: http://visualizingeconomics.com/2006/11/05/2005-us-income-distribution/ Some other mathematical function (perhaps the Gamma distribution http://en.wikipedia.org/wiki/Gamma_distribution) would be a better fit. But the qualitative things I've discussed here would still apply, since you have a distribution that asymptotes to zero as income goes to infinity.)