Using Benford’s Law in Fraud Investigations

Put the numbers to work to boost detection

Posted by Dawn Lomer in Ethics & Compliance, Fraud, Government Program Fraud, Healthcare Fraud, Procurement Fraud, SIU & OIG on July 24th, 2015

Write down your last three addresses, the populations of the four towns closest to you, the cost of gas at the pumps this morning and the amount of your last electricity bill?

Now look at the first digit of each number.

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You’d expect that out of those nine random numbers, the numerals 1 to 9 would be evenly distributed, wouldn’t you? But I bet they weren’t.


A Numbers Game

I’m willing to bet you found far more 1s and 2s than 8s and 9s.

That’s because distribution of first digits in naturally occurring numbers isn’t random at all; it follows a pattern that was discovered by an American astronomer in 1881, who noticed that in logarithm tables, pages that started with 1 were more worn than the other pages.

In 1938, physicist Frank Benford tested the theory on a huge data set from 20 different domains and it became known as “Benford’s Law”.


A Fraud Examiner’s Friend

An auditor applying Benford’s Law has a secret advantage over the unsuspecting fraudster.
“Benford’s Law says that in the natural course of the universe, there’s a certain way that numbers are going to appear – ones are always going to be greater than twos and on down the line,” says Tiffany Couch, founder and principal of Acuity Forensics, a forensic accounting firm based in Vancouver, Washington.

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And this distinct pattern of naturally occurring numbers can be a boon to fraud examiners, she says.


How Does Benford’s Law Work, Really?

Benford’s Law allows fraud examiners to identify outliers in a list of natural numbers by comparing first digits against the probability of their occurrence.

  • The expected occurrence for the numeral 1 as the first digit in a natural number is 30.1 per cent.
  • The expected occurrence for the numeral 2 as the first digit is 17.6 per cent
  • The numeral 3 should be the first digit 12.5 per cent of the time.
  • The probability of the numeral 9 being the first digit in a natural number is 4.6 per cent.

An auditor applying Benford’s Law has a secret advantage over the unsuspecting fraudster.

When numbers are manually inserted into a naturally occurring set the numbers won’t fit the expected pattern and there’s very little a fraudster can do about it.

“It would be very difficult, if not impossible, to get them back into line,” says Couch.


Applying Benford’s Law

While there are many ways to apply Benford’s Law, one of the more common practices is to examine expenditures, Couch explains.

She looks at the records of checks written, electronic funds transfers and other outgoing payments.

“I’ll just run the Benford’s Law on all the expenditure of a company for a certain period of time just to make sure that the ones, twos, threes, fours, fives, sixes, sevens, eights and nines fall in line with what Benford’s Law expects,” she says.

“If it doesn’t, you want to go and track that down.”


Only a Red Flag

Sometimes there’s a logical explanation for numbers that don’t fall in line with Benford’s Law.
“Let’s just say your company has a limit that after $5,000 two people have to sign the check.

A lot of fraudsters will write checks for $4,999, $4,875, or $4,000, not understanding that in the naturally occurring course of the universe they are going to throw [the numbers] out of whack,” says Couch.

“So [in this case] you might see that your fours are out of whack.”


There Might Be a Logical Explanation

That’s an indication that you need to probe further, but it isn’t always an indication that there is fraud occurring. Sometimes there’s a logical explanation for numbers that don’t fall in line with Benford’s Law.

“I audited a school district once where our twos were way more [prevalent] than our ones,” says Couch. “It turned out that all the school teachers got a $250 stipend to start their classroom supplies with and it threw the twos out of whack. So there can be logical explanation but you just have to figure it out.”

A simple, easy-to-apply formula for fraud detection, Benford’s Law is a valuable secret weapon in the fraud examiner’s arsenal.

Dawn Lomer
Dawn Lomer

Managing Editor

Dawn Lomer is the managing editor at i-Sight Software and a Certified Fraud Examiner (CFE). She writes about topics related to workplace investigations, ethics and compliance, data security and e-discovery, and hosts i-Sight webinars.