# Benford’s law

We need to learn to think big.  Humans have had a tendency to underestimate the scale of everything that exists.  We have progressed at an increasing rate from believing the earth was the focus of  existence, to understanding that our planet orbits the sun together with a group of other planets, to appreciating that our sun is a tiny speck in a galaxy that we call the Milky Way that is part of a universe and possibly a multiverse.  We have been able to spot mathematical patterns in nature and to describe them using the equations of physics that in turn allow us to predict the existence of phenomena before we have observed them, such as the Higgs-Boson, and also allow us to harness nature to provide goods and services to society.  The former is the role of physicists and the latter of engineers.  So there is a close link between physicists and engineers and it is not unusual to find engineers working in physics labs and physicists working in engineering organisations.  Frank Benford was a physicist working at General Electric in 1938 when he proposed a law that bears his name, though it has also been credited to Simon Newcomb, an astronomer working 50 years earlier.

Benford’s law predicts the frequency with which the numbers from 1 to 9 will appear as the first digit in a collection of numbers from a real-life source.  The frequency declines logarithmically from 30.1% for 1,  17.6% for 2, 12.5% for 3 etc down to 4.6% for 9.  It is probability distribution so you should not expect see the distribution for every collection of numbers but when it does not appear then you should be suspicious about the provenance of the data, particularly when it does not appear repeatedly.  It is used routinely by accountants and is being used increasingly to identify potential scientific fraud.  Of course some people think big and know about Benford’s law, for instance the fraudster Bernard Madoff filed Benford-compatible monthly returns, which perhaps is one reason why it took so long to catch him.

BTW – Benford’s law does not work for reciprocals or square roots, but is does for powers of 2, factorials and the Fibonacci sequence.

Sources:

Big Bang to Little Swoosh by Max Tegmark, New York Times, April 11th, 2014.

Look out for No.1. by Tim Harford in the Financial Times, September 9th, 2011.

## 1 thought on “Benford’s law”

1. drehack

That Benford law is amazing! The layperson would rather guess an equal distribution for the first digit. And as a challenge: Given Benford’s law holds for a collection of numbers, say for monthly returns in dollars, would this be true as well in pounds at the present exchange rate?

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