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AP Risk: Fun with Numbers

Statistical Law Can Uncover Fraud

Do you know what happened 140, 82, and 25 years ago? No, not other pandemics!  Those are the numbers of years ago a mathematician and statisticians first discovered, then rediscovered and finally proved a statistical law that has fraud-busting application to accounts payable.

The mathematician (and astronomer) was Simon Newcomb. In 1881 he published a paper that showed that in a group of naturally occurring numbers, contrary to assumption, not all numerals (1, 2, … 9) appear with the same frequency in the first digit place. Rather, lower numerals occur more frequently and higher numerals less frequently. In fact, the numeral “1” occurs 30 percent of the time, whereas the numeral “4” occurs about 10 percent of the time, and “9” only 4.5 percent of the time.

Some of you have heard of this before, thanks to the statistician that got involved in our story nearly 60 years after Newcomb. His name is Frank Benford. The phenomenon, called the “law of anomalous numbers” or “significant digit law” is more commonly known as Benford’s Law. In 1938, Benford built on Newcomb’s observations. He demonstrated that the frequency of each numeral appearing in the first digit position, and in the second digit position, of a number decreased inversely with its value. Lower numerals occur more frequently but as the numerals increase, their occurrence as the first (or second) digit decreases.

Fifty-seven years after that, in 1995, Theodore Hill provided a complete proof of the law. If you really want to know, go here (thanks, Project Euclid!).

Here is a table with the frequency of occurrence of numerals in the first two digits:

Why should accounts payable care? Because while you’ve got a lot going on, protecting your organization’s assets is a chief responsibility. And Benford’s law can help. Applied to a collection of vendor invoices, it can point out statistical anomalies that might indicate fraud.

Now there could be reasons for anomalies. If a vendor’s invoice amounts don’t conform to the pattern, it’s not certainly fraud. It’s a flag. You’ll want to take a closer look and be careful not to conclude “fraud” until you’ve investigated.

But as an example, there was a case a few years back in which an analysis showed the numerals 4 and 9 occurring in the first two digits of contract values more frequently than should be the case, according to Benford’s law. The internal auditor took a closer look. It so happened that the company had a threshold of $50,000 at which contracts could no longer be sole-sourced. The higher-than-normal occurrences of contract values of $49,000 – 49,999 were traced back to one contract manager. And it turned out that he was upping contract amounts to just under the $50k limit and sole sourcing them to a company owned by his wife.

Often fraudsters start out carefully, on a small scale. As their efforts go undetected, they get more ambitious and increase the amounts, often to just under some control threshold. But the increase in digits betrays them.

Concerned about the possibility of fraud? Get with your internal auditor and IT and start running checks on contract, invoice and payment amount data. Benford’s Law is considered one of the most powerful techniques for finding fraud.

Did you know that InvoiceInfo provides reporting on user activity, contacts, security, and analysis? For more on invoice-inquiry reporting available from InvoiceInfo, contact us.

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