Power Law

Hundreds of balls are poured on top of a Galton board. Each ball has to randomly bounce off the pegs and drop into one of the bins placed at the bottom. The final pattern of the ball is very predictable and it forms a bell shaped pattern. In statistics this is called as a normal distribution.


How does a randomly bouncing balls form a predictable pattern? If you look into the image carefully the middle bin has lot more balls than the other bins. The reason is because most of the balls will take the same number of leftward and rightward bounces before they reach the bottom. Few balls end up taking more right turns than left and hence they end up some where on the right side bins. Same is true for the balls on the left side bins.

People’s height, weight and IQ are all roughly bell shaped. In the US average adult male height is around 70 inches with a standard deviation of 4 inches and average adult female height is around 65 inches with a standard deviation of 3.5 inches. Normal distributions are useful in knowing the mean and standard deviation.


In normal distributions no importance is given to the outliers. Consider the wealth distribution in United States.

  1. 1% of wealthy Americans have 40% of the country’s wealth. They own 50% of stocks, bonds and mutual funds.
  2. The bottom 80% of the people have 7% of the country’s wealth. The bottom 50% of the people own 0.5% of stocks, bonds and mutual funds.

I am not advocating that US should move towards socialism. The point I am trying to make is normal distribution does not explain this clearly as no importance is given to outliers. Wealth distribution follows a L-shaped distribution. This is called as power law. In power law most values are below average and a few far above which dominate the action. In the power law world

Outliers matter a lot. 

In 2012 there were 634 million websites. Of all these sites the most important ones are those with the most links leading to them. Given below is the graph which plots the number of websites and the inbound links leading to them. Most of the internet traffic is controlled by very few sites. In the power law world

Winner takes it all and rest get nothing.


Power law can be represented by the function of the form

y = C / (x ^ a) 

x = Number of inbound links
y = Number of websites
C is a constant; and the exponent is a

In the book Think Twice – Michael Mauboussin gives few additional examples of power laws

But there are systems with heavily skewed distributions, where the idea of average holds little or no meaning. These distributions are better described by a power law, which implies that a few of the outcomes are really large (or have large impact) and most observations are small. Look at city sizes. New York City, with about 8 million inhabitants, is the largest city in the United States. The smallest town has about 50 people. So the ratio of the largest to the smallest is more than 150,000 to 1. Other social phenomena, like book or movie sales, show such extreme differences as well. City sizes have a much wider range of outcomes than human heights do.

In India millions of kids play cricket. Most of the kids would dream to play for the country. Of those millions only very few get an opportunity to play for the country. Others end up getting nothing. Lots of real world outcomes can be explained by power law.


Power law is also known as the Pareto principle. The Pareto principle (also known as the 80–20 rule) states that, for many events, roughly 80% of the effects come from 20% of the causes. In the image given below you can clearly see that at 20% effort we can get 80% of the results.


In the book Signal and Noise – Nate Silver writes

The name for the curve comes from the well-known business maxim called the Pareto principle or 80-20 rule (as in: 80 percent of your profits come from 20 percent of your customers). As I apply it here, it posits that getting a few basic things right can go a long way. In poker, for instance, simply learning to fold your worst hands, bet your best ones, and make some effort to consider what your opponent holds will substantially mitigate your losses. If you are willing to do this, then perhaps 80 percent of the time your will be making the same decision as one the best poker players like Dwan – even if you have spent only 20 percent as much time studying the game.

If we can achieve 80% accuracy with 20% of effort in poker then do we have something for getting a good outcome in life? It is a very hard question. What if we apply inversion and rephrase the question as What should we avoid so that we do not fail in life. In May 2007 at USC Law school – Charlie Munger did answer this question. Excerpt from his speech

Let me use a little inversion now. What will really fail in life? What do you want to avoid? Such an easy answer: sloth and unreliability. If you’re unreliable it doesn’t matter what your virtues are. Doing what you have faithfully engaged to do should be an automatic part of your conduct. You want to avoid sloth and unreliability.

Here is the video of the speech. Start from 02:44


3 thoughts on “Power Law

  1. Jana,
    Isn’t Power law a mathematical representation of network effect? Meaning, if you look at city sizes as you have pointed out, New York has more inhabitants than ‘smallville’ (ignore the pun!) because of network effect (another variant of ‘positive feedback’). That is, more people, more businesses, more jobs, more migrants, more people. No?

    • Hi Prashanth,

      Yes network effect is one example of Power Law. Very few networks take most of the market share. For example Facebook, Linkedin and Twitter control the entire social networking space. Very few winners but I am sure there would be several other companies in this space which did not get much.


  2. I have a comment about the speaker suggesting not to hold “extreme ideologies” because they block the mind. That depends on who is doing the labeling of “extreme.” For example, at one time abolitionists were seen as having an “extreme” ideology, because the country had normalized slavery. In other words, careful not to confuse being a visionary with being an ideologue. For example, 20 years ago, while doing my coursework for a masters degree in political philosophy, I pointed out in my Democratic Theory seminar that ancient Greece was not a democracy, but an elite oligarchy because only propertied free males could vote. I was laughed out of the room for my “extreme” views by the other (all male) students and viewed as having a bias because I am female. Of course, I was the one without bias pointing out the bias that was invisible because it was so normalized. Today, that same course, taught at the same school, accepts as given that ancient Greece was not a true democracy, according to the definition of the word.

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