Rule of one
Benford's law has been used to detect 
fraud in tax, voting, and accountancy, since
doctored numbers tend not to follow the
Image: E_Y_E/iStockphoto

Amid the swirling chaos of our world an almost perfect pattern surrounds us. It connects random events and phenomena – from the large to the small, and the monumental to the inconsequential.

It can be seen in the spluttering spins of dying stars and the numbers in our electricity bills. You can glimpse it in the dizzying heights of towering skyscrapers and in the darkest depths of earth-shaking quakes. It’s in the time it takes a burst from a gamma ray to reach your eye, as well as numbers randomly picked from newspapers. Death rates, stock prices, population sizes, baseball statistics, the Fibonacci sequence; in all of these, one thing remains constant. Numbers with larger first digits are less likely to occur than smaller ones - and not a bit less likely, but significantly so.

Professor Malcolm Sambridge from the Research School of Earth Sciences says that this unexpected quirk in nature was identified by an obscure mathematical rule first discovered in the 19th century and then quickly forgotten. Today the law, known as Benford’s, is revealing even more remarkable relationships in the physical sciences.

Benford’s law says that the digits of numbers related to natural events are not uniform but distributed in a specific way. Much to many people’s surprise, the natural world is littered with a surplus of numbers starting with the digit one.

“Most physicists would think that the likelihood of a number beginning with a one would occur just as often with numbers beginning with a two or a three, or so on,” says Sambridge.

“But it turns out that this is not the case in the natural world.

“Instead, as the theory shows, roughly 30 per cent of numbers related to many real-world events begin with the number one and only 17 per cent begin with a two. And it goes right down to roughly about four per cent beginning with a nine,” he adds.

It even applies to street addresses.

“If you select 20 friends and you take the first digit of their door numbers, you will find there are roughly twice as many that begin with four and below than five and above,” explains Sambridge.

Now Sambridge, working with ANU colleague Dr Hrvoje Tkalčić and Professor Andrew Jackson from ETH Zürich, has vastly expanded the range of natural phenomena shown to conform to Benford’s law.

“For many years Benford’s law was a mere mathematical curiosity,” says Sambridge. “However, today it can be applied more widely than previously ever imagined.

“And whilst applications of the Law have increased in recent times, there have been few in the physical sciences. So we decided to see if Benford’s law could be observed in the fields of physics, astronomy, geophysics, climate science, medicine and biology.”

The researchers tested 18 data sets comprising over 750,000 numbers across a range of phenomena that included greenhouse gas emissions, the masses of giant planets outside our solar system, and the number of infectious diseases reported to the World Health Organization. They even looked at total career runs scored by cricket players in test matches between 1877 and 2010.

“Much to our surprise, we found that Benford’s law largely holds true in all these areas,” says Sambridge.

And the surprises didn’t stop there. The researchers even managed to detect, for the first time ever, a physical phenomenon – discovering an earthquake which took place in Canberra during the 2004 Asian tsunami.

“One of the things we are interested in is automated methods of detecting an earthquake and we found that Benford’s law can be used for exactly that purpose,” says Sambridge.

“Essentially, we are taking the first digits of the counts of a seismometer, which measures motions in the ground caused by earthquakes. We then discard the rest of the information.”

So unexpected is Sambridge’s application of Benford’s law that when it is explained to other scientists it often meets with sheer disbelief.

“Ultimately, I think the whole idea captures the imagination, because most people haven’t heard of this idea and it comes as a surprise that it would be true, but it is,” says Sambridge.

“It’s early days, really, but it’s something we’re excited by because it’s a completely new way of looking at many different kinds of problems.”

So look around you. Is one the loneliest number or is it in good company in the digits that surround you?

Benford’s law in one, two, three

Benford’s law has already been used to detect tax, voter and accountancy fraud as well as digital image manipulation. This is because doctored numbers tend not to follow the rule.

In addition, researchers have also identified Benford’s law in some unusual places throughout the physical sciences.

One: Global temperature anomalies between 1880 and 2008.

Two: The strength of the earth’s gravity.

Three: How many times the earth’s magnetic field has flipped from the North to the South Pole.


Editor's Note: A story provided by ANU Reporter.  This article is under copyright; permission must be sought from ANU to reproduce it.