Physicist Frank Benford first popularized Benford’s Law in 1938. In the fight against fraud, however, it’s a statistical concept that remains relevant even today. It was even cited as an effective tool for business to detect fraud early on at a recent PayrollOrg Leaders Conference session on payroll analytics. A version of Benford’s law is also utilized by the IRS to detect falsified tax data.

The Most & Least Popular Numbers

Benford’s Law is simple: In sets of naturally occurring data, multidigit numbers beginning with one, two, or three generally are more likely to occur than those starting with four through nine. Indeed, studies have determined that numbers beginning with one will occur about 30% of the time, and numbers beginning with two will appear about 18% of the time. In contrast, numbers beginning with nine will occur less than 5% of the time.

These probabilities are considered both “scale invariant” and “base invariant,” meaning the numbers involved could be based on, for example, the prices of stocks in dollars, yen, or euros. As long as the set includes at least four numbers, the first digit of a number is more likely to be one than any other single-digit number.

Best Uses For The Rule

This rule has important implications for fraud investigations. To avoid raising suspicion, fraud perpetrators often use figures they believe will replicate randomness. Usually, they choose a relatively equal distribution of numbers between one and nine. Such numbers don’t only appear in fraudulent payroll records, of course. The same principle applies to fudged tax returns, inventory records, expense reports, accounts payable or receivable, general ledgers, and other financial documents.

Forensic accountants typically use software informed by Benford’s Law and other analytic principles to uncover anomalous numbers. But that’s only a starting point. Most investigations involve such elements as computer forensics, interviews with suspects and witnesses, and discussions with business owners and their legal counsel. In some cases, anomalous numbers have innocent explanations.

When It Doesn’t Work

Benford’s Law itself isn’t infallible. It may not work with smaller sets of numbers that don’t follow the rules of randomness or numbers that have been rounded (resulting in different digits). Smaller numbers are more likely to occur simply because they’re smaller and the logical place to begin a count.

Assigned numbers, such as those on invoices, may also break the rule. And uniform distributions — such as lotteries where every number painted on a ball has an equal likelihood of selection — may not suit a Benford’s Law analysis. Further, prices involving the numbers 95 and 99 (often used as part of a marketing strategy) may call for a different approach.

Try Basic Searches

Fraud professionals typically use sophisticated analytics (including AI) to examine large volumes of data. But Benford’s Law can also be applied to smaller sets with basic spreadsheet software. If you spot numbers that appear to violate the rule when reviewing your company’s financial records, consult your attorney and turn the case over to a forensic accountant. Contact us for help or more information.