When it comes to fighting fraud, there’s a tried-and-true statistical precept that remains as relevant and widely accepted as ever. “Benford’s Law” is often used by forensic accountants to spot dubious digits — and catch even the most sophisticated thieves.
Historical Background
The principle is named for Frank Benford, a physicist who noted in the 1930s that, in sets of random data, multi-digit numbers beginning with 1, 2 or 3 are more likely to occur than those starting with 4 through 9. 
Studies have shown that numbers beginning with 1 will occur about 30 percent of the time, and numbers beginning with 2 will appear about 18 percent of the time. Those beginning with 9 should occur less than 5 percent of the time.
These probabilities have been described as both “scale invariant” and “base invariant,” meaning the numbers involved could be based on, for example, the prices of stocks in either dollars or yen. As long as the set includes at least four numbers, the first digit of a number is more likely to be 1 than any other single-digit number.
Striking Implications
Benford’s Law carries striking implications for fraud detection. To avoid raising suspicion, fraud perpetrators often use figures they believe will replicate randomness. Typically, they choose a relatively equal distribution of numbers beginning with 1 through 9 in the mistaken belief that all nine digits are equally probable.
Fraud investigators can take advantage of these errors and test data in a variety of financial documents such as:
- Tax returns,
- Inventory records,
- Expense reports,
- Accounts payable or receivable,
- General ledgers, and
- Refund reports.
Although software programs based on Benford’s Law exist to examine massive amounts of data, the principle is simple enough to apply using spreadsheet software. For example, Excel can analyze random groups of numbers by determining the distribution of the first digits of those numbers by building a table with rows for each digit (1-9) and columns for the:
- Frequency with which numbers beginning with each digit occur in the random sample,
- Percentage rate of that frequency, and
- Percentage rate of the frequency to expect according to Benford’s Law.
Spreadsheet software can easily convert the table into a chart that graphically illustrates any significant discrepancies between the actual and expected occurrence of the first digits. A chart that shows too many numbers beginning with 9 and too few with 1 should raise red flags and prompt further investigation.
Not Infallible
Benford’s Law, however, isn’t infallible. The law may not work in cases that involve smaller sets of numbers that don’t follow the rules of randomness or numbers that have been rounded (resulting in different first digits). Also, smaller numbers (1, 2, etc.) 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, are also iffy. On a similar note, 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. And prices involving the numbers 95 and 99 (often used because of marketing strategies) may call for a different approach.
In addition, the principle may be ineffective for sets of numbers with built-in ceilings and floors. For example, expense reports where receipts are required for meals costing $25 or more will reveal many claims just under the limit, in amounts such as $24.90.
Foundational Approach
Benford’s Law isn’t appropriate in every instance. And, as advanced metrics continue to forge new inroads into fraud detection, it may take a backseat to new methods. But for now, Benford’s Law remains a foundational approach to finding fraud through statistical analysis.
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