ALEX HAD NO IDEA what dark little secret he was about to uncover when he asked his brother-in-law to help him out with his term project. As an accountancy student at Saint Mary's University in Halifax, Nova Scotia, Alex (not the student's real name) needed some real-life commercial figures to work on, and his brother-in-law's hardware store seemed the obvious place to get them.
Trawling through the year's sales figures, Alex could find nothing obviously strange about them. Still, he did what he was supposed to do for his project, and performed a bizarre little ritual requested by his accountancy professor, Mark Nigrini. He went through the sales figures and made a note of how many started with the digit 1. It came out at 93 per cent. He handed it in and thought no more about it.
Later, when Nigrini was marking the coursework, he took one look at that figure and realised that an embarrassing situation was looming. His suspicions hardened as he looked through the rest of Alex's analysis of his brother-in-law's accounts. None of the sales figures began with the digits 2 through to 7, and there were just 4 beginning with the digit 8, and 21 with 9. After a few more checks, Nigrini was in no doubt: Alex's brother-in-law was a fraudster, systematically cooking the books to avoid the attentions of bank managers and tax inspectors.
It was a nice try. At first glance, the sales figures showed nothing very suspicious, with none of the sudden leaps or dives that often attract the attentions of the authorities. But that was just it: they were too regular. And this is why they fell foul of that ritual he had asked Alex to perform.
Because what Nigrini knew - and Alex's brother-in-law clearly didn't - was that the digits making up the shop's sales figures should have followed a mathematical rule discovered accidentally over 100 years ago. Known as Benford's law, it is a rule obeyed by a stunning variety of phenomena, from stock market prices to census data to the heat capacities of chemicals. Even a ragbag of figures extracted from newspapers will obey the law's demands that around 30 per cent of the numbers will start with a 1, 18 per cent with a 2, right down to just 4.6 per cent starting with a 9.
It is a law so unexpected that at first many people simply refuse to believe it can be true. Indeed, only in the past few years has a really solid mathematical explanation of its existence emerged. But after years of being regarded as a mathematical curiosity, Benford's law is now being eyed by everyone from tax inspectors to computer designers-all of whom think it could help them solve some tricky problems with astonishing ease. In two weeks' time, the US Institute of Internal Auditors will begin holding training courses on how to apply Benford's law in fraud investigations, hailing it as the biggest advance in the field for years.
The story behind the law's discovery is every bit as weird as the law itself. In 1881, the American astronomer Simon Newcomb penned a note to the American Journal of Mathematics about a strange quirk he'd noticed about books of logarithms, then widely used by scientists performing calculations. The first pages of such books seemed to get grubby much faster than the last ones.
The obvious explanation was perplexing. For some reason, people did more calculations involving numbers starting with 1 than 8 and 9. Newcomb came up with a little formula that matched the pattern of use pretty well: nature seems to have a penchant for arranging numbers so that the proportion beginning with the digit D is equal to log10 (1 + [1/D]).
With no very convincing argument for why the formula should work, Newcomb's paper failed to arouse any interest, and the Grubby Pages Effect was forgotten for over half a century. But in 1938, a physicist with the General Electric Company in the US, Frank Benford, rediscovered the effect and came up with the same law as Newcomb. But Benford went much further. Using more than 20 000 numbers culled from everything from listings of the drainage areas of rivers to numbers appearing in old magazine articles, Benford showed that they all followed the same basic law: around 30 per cent began with the digit 1, 18 per cent with 2 and so on.
The article in full is at:
http://www.newscientist.com/ns/19990710/thepowerof.html
It turns out that the reason for Benford's Law has something to do with the ULTIMATE distribution, which is got by taking chunks from various probability distributions (eg uniform, normal, poisson) and putting them together. The law, which has already been used to uncover fraudulent activities worth millions of dollars, isn't limited to crime fighting - it has also been used to allocate computer disk space more optimally and to provide a quickcheck on simulations.
- Robert Matthews