Number ‘one’ theory proves that numbers can’t lie.

Forensic accountants are increasingly using data analytics in identifying fraud. Many data analysis tools are available, however ‘Benford’s Law’—also known as ‘First Digit Law’—is used regularly. Before we demonstrate how Benford’s Law applies in a forensic case, first we need to explain, what is Benford’s Law?

Named after physicist Frank Benford in 1938, however Simon Newcomb, a Canadian mathematician, had previously documented the law in 1881. Effectively the law states that in a list of numbers, more numbers should start with a one than any other digit, followed by those that begin with two, then three, and so on. “The low digits are expected to occur far more frequently than the high digits”, says Mark J. Nigrini, author of Benford’s Law: Applications for Forensic Accounting, Auditing and Forensic Accounting.

According to Benford’s Law, the digit one should account for 30% of leading digits and each successive number should represent a progressively smaller proportion with nines coming last at under 5%.

Mathematically, Benford’s Law is represented as:

The table below shows the expected distribution of digits according to Benford’s Law:

As an experiment, I have obtained the following graph from the website www.datagenetics.com, which plots the frequency of the first digit in the altitude of the top 122,000 most populated towns in the world.

You can see how the data follows, almost exactly, the expected results according to Benford’s Law.

So, how do forensic accountants use Benford’s Law to assist with identifying fraud?

Where to look first can be difficult, for a forensic accountant in identifying whether fraudulent activities were present. For example, a large accommodation booking agency engaged forensic accountants as its national call centre was issuing refunds and something was not adding up. When the accountants started analysing the data, they found a large volume of refunds between $40 and $59 (i.e. amounts starting with a 4 or a 5). According to Benford’s Law, it is expected that more refunds would start with a one; however this was not the case.

The call centre staff could issue up to $60 in refunds to customers—anything over $60 required a supervisor to authorise it. The accountants found a high percentage of refunds beginning with a 4 or a 5, which led them to think that refunds were being issued close to the $60 threshold. By tabulating the refund data the accountants’ suspicions were confirmed, and ultimately uncovered a few call centre staff were fraudulently processing refunds to themselves, their family and friends, amounting to hundreds of thousands of dollars.

Benford’s Law is only one approach and it is not appropriate for all data sets. But it is a good tool to assist in identifying data anomalies, which further investigation must explain.