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INSIGHT – One fraud easily spotted.

Here’s a question for the more numerate among you: what proportion of any large ragbag of numbers – your sales invoices, for example, or the drainage area of rivers in hectares – will begin with the digit “1”? The logical answer is, of course, 11.1% (one-ninth, assuming that such numbers don’t begin with zero!). The right answer, however, is around 30%, says a report in the 10 July issue of New Scientist. Confused? A long-forgotten law of numbers reveals that numbers are much more likely to begin with a 1 – and least likely (just under 5% probability) to begin with a 9. (The formula to calculate the frequency of appearance for any digit is given in the box.) This almost-inexplicable phenomenon, which is known as Benford’s Law, seems to be one of those weird mathematical laws that are found in nature. Another example is the Fibonacci sequence – 1,1,2,3,5,8,13, …, in which each number is the sum of the preceding two. It can be found in countless places: the structure of plant leaves and pineapple skins being just a couple of instances. New Scientist reports that Benford’s Law was first discovered by a 19th century astronomer who’d noticed that the first pages of books of logarithms tended to be much more heavily thumbed than the later pages: people did more calculations involving numbers beginning with 1 or 2 than with 8 or 9. In 1938, Frank Benford, a physicist at US General Electric rediscovered the phenomenon and demonstrated that it holds true in thousands of instances. But why – and so what? The “why” is difficult to answer, but it appears to work with collections of numbers that are not utterly random, nor rigidly constrained (prices of competing brands of beer, for instance). There’s a useful purpose to all this, however, and it’ll make FDs’ eyes pop out: your accounts books probably adhere to Benford’s Law and, if they don’t, there may well be some fraudulent activity going on. Mark Nigrini, an accountancy professor at Southern Methodist University in Dallas, used the mathematical law to expose shady dealings at a hardware store run by the brother-in-law of one of his students. Nigrini had asked him to count how many of the sales figures began with each of the nine digits. Instead of 30% beginning with 1s, 93% did. The accounts were “too regular”. Using what is now known as “digital analysis”, one American company has already spotted frauds worth more than $1m. In one of them, too many cheques were being illicitly issued to the fraudster for amounts ranging between $6,500 and $6,599. Auditors are also picking up on the use of Benford’s Law, according to Nigrini. “Ensuring (that a) fraud always complies with Benford’s Law is going to be tough,” he told the magazine, “and most fraudsters aren’t rocket scientists.”

 YOUR NUMBER'S UP The formula that determines how frequently a number starts with a particular digit D is given by: log10 of 1+(1/D). Here's the pattern: By the way, 46% of the 70 numbers on these two pages begin with a 1. Blame FRS10. 1st digit     freq. 1             30.1% 2             17.6% 3             12.5% 4              9.7% 5              7.9% 1st digit     freq. 6              6.7% 7              5.8% 8              5.1% 9              4.6%

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