It’s lurking in everything from company accounts and stock market data to tax returns and even the prices paid for racehorses.
It is the bizarre law governing the frequency of seemingly random numbers - which is now seen as a crucial weapon in the global fight against fraud.
It’s called Benford’s Law, and it predicts that in a large collection of data, around 30 per cent of the numbers will start with the digit "1".
First discovered over 130 years ago, its existence has long provoked controversy.
Earlier this month researchers showed how Benford’s Law could help combat criminal activity in the USD 10 trillion-plus international trade business.
Such hard-nosed applications belie the law’s humdrum origins. During the 1880s, an American astronomer noted that books of logarithms, then widely used to carry out complex calculations, seemed grubbier near the beginning. For some unknown reason, it seemed that calculations more often involved numbers starting with the digit “1” than any other.
There even seemed to be a rule of thumb governing the phenomenon, according to which 30 per cent of calculations involved numbers begin with the digit "1", around 18 per cent start with a "2", 13 per cent with “3”, down to just 5 per cent with the digit "9".
For decades, this bizarre discovery was dismissed as just a curiosity. Then in 1938 a physicist at the US company General Electric named Frank Benford found the same proportions of digits among tens of thousands of numbers extracted from sources as diverse as articles in Reader’s Digest to geographical data.
Yet despite confirming the reality of the “law”, Benford couldn’t explain why it worked with so many different sources of numbers. And in the absence of a solid explanation, suspicions remained that it was just some kind of fluke.
Not until the 1990s did glimmerings emerge of a theory underpinning the law. Roughly speaking, the idea is that collections of data should have the same properties no matter what units they’re measured in.
So, for example, the ranking of race-horses auction prices should be the same whether they’re given in dollars, dirhams or sterling. Similarly with, say, listings of the areas of Asian countries or US stock market prices. The data-sets should reveal the same insights whether they’ve been collected by humans today or aliens a thousand years from now; the units should be irrelevant.
It turns out there’s only one way of ensuring such a common-sense feature of data – and that’s if the leading digits follow Benford’s Law.
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With confidence growing that it’s not just a fluke, the law is being increasingly used to spot anomalies in data-sets – and to reveal potential fraud.
Pioneered by Professor Mark Nigrini at West Virginia University, the technique involves analysing the raw numbers in, say, company accounts and checking that around 30 per cent of them begin with “1”s, 17 per cent begin with “2”s, all the way down to 5 per cent beginning with “9”s.
If they don’t, it can be because someone has been cooking the books.
For example, Prof Nigrini has argued that the financial chicanery that led to the collapse of the US energy company Enron in 2001 could be seen in deviations from Benford’s law in the company’s accounts.
Other researchers have since used the law to uncover evidence that during recessions many companies amend their financial statements to make them look better.
Even official statistics issued by nations have fallen under suspicion after analysis using Benford’s Law.
In 2011, audit experts in Germany reported finding large deviations from the expected frequency of digits in economic reports submitted by Greece to EU regulators – seemingly confirming suspicions raised by the European Commission that the government was manipulating the raw data.
Now a team led by Prof Andrea Cerioli at the University of Parma, Italy, plan to use Benford’s law to tackle a truly global problem: fraud in international trade.
But to do it, they have had to address concerns about the reliability of the law. Critics have pointed out that not all data sets conform to its predictions – and it’s hard to tell whether discrepancies really are evidence of fraud or just flukes.
That raises the risk of wrongful accusations – undermining confidence in the technique.
Prof Cerioli and his colleagues have tackled the problem by simulating the kind of data expected from legitimate trade transactions. That gives a benchmark for the size of deviations expected even when there’s no fraud – thus making evidence of shady behaviour more reliable.
The team has now applied their technique to real-life trading data from the European Union. In results published earlier this month in the US journal, Proceedings of the National Academy of Sciences, they show that their methods can both detect fraudulent activity and exonerate legitimate data which doesn’t perfectly match the law’s predictions.
The team has now set up a website allowing customs officials to carry out checks using the new technique.
The approach used by Prof Cerioli and his colleagues is likely to add more confidence to the use of Benford’s law – and to encourage more applications.
Even so, Prof Cerioli and his colleagues admit that no technique can guarantee picking up truly determined fraudsters. As word spreads about the use of Benford’s law by officials, so will efforts to undermine its power.
Just ask Bernard Madoff, who operated his notorious $65 billion Ponzi scheme until he was unmasked in 2008.
An analysis of his investment “trades” – which notoriously never actually took place – showed some deviation from Benford’s law, though nothing spectacular. But court papers later suggested that Madoff was aware his records might be subjected to forensic analysis, and used techniques to evade them.
Whether he knew about Benford’s law isn’t clear. And it’s hard to find out, for despite his ingenuity Madoff ended up in jail, and is not due for release for another 120 years.
Robert Matthews is Visiting Professor of Science at Aston University, Birmingham, UK
