The modern world is full of numbers — bank bailouts, government debts, MPs’ expenses — and it is only human to find them confusing and perhaps, at times, even suspicious. But it is hard to check so many numbers directly. An alternative is to look for patterns in the data that might indicate that something is amiss.
One such pattern is Benford’s Law, named after a physicist at General Electric. In 1938, Benford analysed 20,229 sets of numbers including such wildly different categories as the areas of rivers, baseball statistics and all the numbers collected from an issue of the Reader’s Digest. He found something surprising: numbers whose first digit was one cropped up 30 per cent of the time, much more than we might expect.
Benford’s Law does not apply to every set of numbers — for example, it does not apply to post codes or national insurance numbers, which are assigned by bureaucratic processes. But all sorts of “natural” processes should produce Benford data.
To see why this might happen, imagine a $10bn investment fund, growing at a steady 10 per cent a year. It must grow by 100 per cent before the first digit changes, to $20bn. This will take just over seven years. By the time the fund has grown to $50bn, only 20 per cent growth is needed to reach the next digit at $60bn; this will take less than two years. To grow from $90bn to $100bn takes just over a year, but to change the first digit back to two again, at $200bn, requires another seven years. Such processes seem to explain why many numbers begin with the digit one, and only one in 20 begin with the digit nine.
Since the 1990s, Benford’s Law has been used to uncover dodgy accounting, frauds and tax evasion — for example, inventing invoices that are just under the threshold for managerial approval.
Mark Nigrini, an accountancy professor, is a leading practitioner, while Hal Varian, now chief economist at Google, once argued that Benford’s Law is a good test of an economic model. If data satisfy Benford’s Law on the way in, but not out, time to ask questions about the model.
Still, no test is foolproof. Bernard Madoff, who last year was sentenced to 150 years in prison for a massive fraud, would have passed a Benford’s Law test, according to Paul Kedrosky, a financial commentator who analysed fifteen years of Madoff’s monthly returns. US regulators now believe that Madoff’s team deliberately used a randomising program to make the numbers seem valid. Even Benford’s Law can be evaded — if a fraudster is sophisticated enough.
Tim Harford, a Financial Times columnist, is author of Dear Undercover Economist
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