Power Law

A Power Law curve.

Take a look at the image to the right. It is very likely the most important chart they never taught you about in school. Failing that, it’s almost certainly the most important chart that you may have seen from time to time and learned a bit about but never realized the significance of.

What is it? Depending upon how you label the axes, it could represent a whole host of things. In fact, it’s kind of shocking just how many things in the real world this curve represents. Depending upon the dataset, you may have to flip the curve.

It goes by several names: Exponential distribution. Logarithmic function. But one of the most popular names, and perhaps the most ominous, is the Power Law curve. From Wikipedia:

In statistics, a power law is a functional relationship between two quantities, where one quantity varies as a power of another.

Or, more simply, it’s what you get when you plot out a function that looks like this:

f(x) = axk

Normal Distribution – the “Bell Curve”

The name “Power Law” comes from the k – the “power” or exponent in the function. If you’ve had any introduction to statistics at all, you’re probably much more familiar with the chart to the left – the famous “Bell Curve,” or, more accurately, the Normal Distribution. Unless you took an actual “full” statistics course from the math department (many science majors these days get away with “statistics for <insert your department> majors” instead of taking the one from the actual math department), you probably didn’t spend enough time studying it. Even if you did take a “real” statistics course, you probably didn’t fully appreciate the significance of it. Don’t feel bad. It’s very likely that your professor didn’t appreciate it, either.

Most people are taught that the Normal Distribution is the most common distribution that you find in nature. Indeed, this is why it’s called the “normal” distribution. It’s very common. Quite a few people, however, mistakenly believe that it’s the only distribution you find in nature – that everything follows a bell curve distribution.

The second variant is patently false. Quite a few things follow other distributions (these are not the only two you will find; there are quite a few others). The first formulation, though, isn’t quite accurate either. Yes, a lot of things follow normal distributions. But quite a lot of things don’t. In fact, the prevalence of the normal distribution is actually what leads to so many cases of the power law distribution.

Because there are some key things about the power law distribution that you were never taught. Here are a few:

Power Law Distributions arise from iteration of Normal Distributions

OK, what does that mean? Let’s consider the following very generic set of circumstances:

  1. A competitive event.
  2. The population of competitors is unequal
  3. The inequality is distributed along something resembling a normal distribution.
  4. Winners from any given round of competition keep their winnings.
  5. The winnings form any round confer an advantage in subsequent rounds.
  6. Competition is iterated over multiple rounds.

Whenever these six conditions are met, after many rounds of competition the results will always form a power law curve. Always. Without exception.

With Power Law Distributions, Averages are Meaningless

Not just averages. The mean, median and mode are all meaningless in a power law distribution. They literally tell us nothing. We are taught as early as middle school to use these numbers to analyze large datasets. Indeed, for many of us they are the only ways we know to get meaning out of those sets. But for anything that follows a power law curve, they literally tell us nothing about the data. Well, OK, not quite nothing:

If the mean (average) and the median are wildly different, that’s a strong hint that the data actually follows a power law curve. In a normal distribution, they will be very close together. In a power law distribution they might be, but they probably won’t be. [The converse is not true: a power law distribution doesn’t necessarily have wildly different medians and means.]

What it all Means

The Power Law curve, combined in some cases with other important information, is the iron law of mathematics defining why:

  • Income and wealth inequality are an inescapable part of the human condition.
  • Top members of some professions (sports, film, tv, music, writing, journalism, and many others) make millions while nobody else can even make enough to live on.
  • Globalism is very bad for everyone except those at the very top.
  • Current levels of immigration in the US are too high.
  • Tenure is bad.
  • Big corporations are terrible.
  • Big government is worse.
  • Most mature industries coalesce around a very small number of very large firms.

And much, much more. But more on those another day.

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