The mathematical law that shows why wealth flows to the 1%

One of the main issues raised by the Occupy demonstrators is the inequitable distribution of wealth. Their slogan focuses on the extreme difference between the richest and the poorest: "We are the 99%," say the banners and T-shirts, pointing out that 1% of the world's population has somehow clawed its way to disproportionate money and power. Time to do something about this unnatural distribution, no?

The economist Edward N Wolff, of New York University, has pointed out that, as of 2007, the top 1% of households in America owned 34.6% of all privately held wealth, and the next 19% had 50.5% of the wealth. This means that just 20% of the people owned 85% of the wealth, leaving only 15% for the bottom 80% of the people. No one who is interested in an equitable society can fail to be irked by this unfairness.

But the unfairness is, unfortunately, not unexpected. What the protesters are fighting (consciously or unconsciously) is the 80/20 rule – variously called Pareto's principle, Zipf's law, the long tail or Benford's law, depending on what you are studying – a staple in scientific, economic and business textbooks, the go-to idea to show how the frequency of a set of natural events is not always what you might recognise as, well, natural.

The maths underlying the 80/20 rule, known as the power law distribution, is found in many natural systems over which no single human has much influence. Its concentration of the extremes seems built into the fabric of complex systems that depend on numerous factors that continually change over time.

The simplest version says that 80% of your company sales will come from 20% of your customers; that 80% of the world's internet traffic will go to 20% of the websites; 80% of the film industry's money gets made by 20% of its movies; 80% of the usage of the English language involves just 20% of its words. You get the picture.

A distribution based on a power law says extreme events (or richest people, or biggest websites) account for most of the impact in that particular world, and everything falls off quickly afterwards. The combined wealth of the top 10 richest people in the world is orders of magnitude greater than the next 10, which is orders of magnitude greater than the next 10, and so on. The rest of the field sits in a long, almost-irrelevant tail.

This distribution might sound odd. At school, we're introduced to a different distribution, the more familiar "normal" (or Gaussian), which is best displayed in the bell-curve spread of values around an average. Measure the heights of a random selection of men, say, and most will be around the average value, with progressively fewer as you go in either direction away from the middle. Plot this on a graph and you get the bell curve.

Power law distributions, however, do not cluster around a single value. The impact of one big earthquake, for example, is bigger than the sum of millions of smaller, more common ones. Very few huge solar flares erupt from the surface of the sun, but those few are more significant than the endless thousands of smaller ones. The same applies to the numbers of big cities, the size of the Moon's craters and the occurrence and citations of scientific papers.

Once you know power law distributions exist, they become very useful. The concept of the "average" is useless, for example, when talking about things that follow power laws. The average height of the people in a room (following the normal distribution) might tell you a lot about the spread heights of people in that room, but the average wealth of a country's citizens (which follows a power law distribution) tells you little or nothing about how rich or poor most people are. And listening to the maths also tells you that the Occupy protesters have got it right that focusing on the extremes (a tax on the wealthiest 1%, say) will bring disproportionate results for the number of people it will affect.

Ben Goldacre's Bad Science column will return next year