From a Normal Distribution to a Pareto

A model to illustrate income inequality

In last week's post, “The Normal Density and Percentiles,” I showed how the distance between percentiles in a normal distribution grows as we move toward the tails. Now, normal distributions are important, but there are other distributions that are just as important but aren't talked about in a standard statistics course. In this post, I'll show how a normal distribution can lead to a Pareto distribution.

Most scenarios that come from averages fit the normal distribution. Height is a good example because we could say that a person's height is an average influenced partly by the heights of their parents, grandparents, etc., and partly by their environment. It is also a fair assumption for skills and ability. Now, let's say that our community has a normal distribution of skills and that the average hourly wage is $20. We'll pay you an extra $10 per hour for every standard deviation from the mean, with a $10 per hour minimum. In this case, the distribution shown in Figure 1 was made by randomly picking 100,000 people from a normal distribution (which is why the median isn't exactly $20).

Figure 1: The distribution of pay if the median person earns $20 per hour, pay is increased by $10 per hour per standard deviation from the mean in a normal distribution, and the minimum pay is $10 per hour.

Figure 1 is, as expected, skewed to the right. With some people making at least $60 per hour. 33% of all income goes to people who make more than the 80th percentile. In the real world, pay goes up faster than a straight line as skills go up. Sports is the best example of a situation where the best player makes a lot more than the average player. To illustrate this, we'll change the way pay is scaled.

In our second case, pay goes down by $10 per standard deviation below the mean, but it goes up by (exp(x)-1)*10 per standard deviation above the mean, where x is the number of standard deviations from the mean. In other words, it will grow as an exponential function times $10, where we subtract 1 so that it starts at 0 (remember that exp(0)=1). Again, nobody earns less than $10 an hour. Figure 2 shows the distribution of the results. We stopped the x-axis at $100, but the highest hourly pay is $777.63. Look at how the top 20% of earners now get 44% of the total income.

Figure 2: The hourly pay when pay decreases by $10 per standard deviation below the mean but increases by (exp(x)-1)*10, where x is the number of standard deviations from the mean per standard deviation above the mean.

In this context, a Pareto distribution would have the top 20% earning 80% of the income. The Pareto distribution is actually a family of distributions, and the “80-20 rule” only applies to those with a shape parameter of about 1.16. We're not there yet, so let's change our pay scale so that people who make more than the mean get an extra (exp(x-1)*60). In this case, the only difference is that we increased by $60 instead of $10. Why $60? The point is to keep the example as easy as possible, and $60 works well enough. What we get is shown in Figure 3.

Figure 3: This is the same as Figure 2, but we scale by (exp(x)-1)*60.

The top 20% of earners now make about 70% of the money, with a maximum pay of $4565.79. The way we set up the pay scale isn't very smooth because it drops off at $20. Still, this gets the point across. In a situation like this, what would a Perato distribution look like? Figure 4 shows a Perato distribution with a minimum value or shape value of $10 and a scale factor of 1.16. The 80th percentile is lower, but the middle is about the same. On the other hand, the maximum is much higher at $38944.23, so the tail is much longer.

Figure 4: A Pareto distribution with a scale factor of 1.16 and a minimum or shape value of $10.

Pareto distribution, or really the '“80-20 rule,” fits a number of natural phenomena, such as the distribution of the size of wildfires or the strength of earthquakes, as well as human activities. Here is a great list and a few examples from that list: 80% of the wealth is owned by 20% of the population, 80% of crimes are committed by 20% of criminals, 80% of your knowledge is used 20% of the time, 80% of a company's absenteeism is caused by 20% of staff, and an inverse example is that 20% of your wardrobe is worn 80% of the time.

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