Investigators and internal auditors are using a somewhat simple mathematical principle to catch fraudsters. Known as “Benford’s law,” it is the idea that the distribution of digits in multi-digit natural numbers is not random, nor is it equally distributed; instead, it follows a predictable pattern.
In the realm of data analysis, Benford’s Law stands as a remarkable phenomenon. Named after physicist Frank Benford who discovered it in 1938, this mathematical principle has found applications in a diverse range of fields, from forensic accounting to fraud detection. Internal auditors conducting data analytics can use it as an easy check on data sets to look for fraud and manipulation.
Understanding Benford’s Law
Benford’s Law asserts that in many naturally occurring datasets, the leading digits tend to follow a specific distribution. Surprisingly, the number 1 appears as the leading digit approximately 30 percent of the time, followed by 2 with around 18 percent, and so forth, forming a logarithmic pattern. Conversely, larger digits like 8 and 9 occur as leading digits less frequently, at around 5 percent and 4 percent respectively.
For example, let’s say you have a three-digit payment amount ($XXX). What are the odds that the first digit would be 3? Well, you might think the odds are 1 in 9, or that there is an equal chance that the first digit would be 1, 2 or 3, up to 9. But that’s not the case. Benford’s Law says that the odds are not random. In fact, there is a 12.5 percent chance the first digit will be a 3. There is a 30 percent chance the first digit will be a 1.
Fraud examiners can use Benford’s Law to test natural numbers, like payment amounts and ensure they follow the expected outcomes. Human generated numbers don’t always follow Benford’s law, so the theory is that if a fraudster submits fake invoices for payment, those numbers won’t follow the expected distribution of the digits. If done enough times, it upsets the natural order of the way numbers should occur (according to Benford). For example, if you run a Benford’s Law test on last month’s payments, and you find the first digit was 8 in 32 percent of the payments, that’s an anomaly. Bendford’s Law says 8 should be the first digit only 5 percent of the time.
The underlying principle is grounded in the notion that human-generated data is often not uniformly distributed. It emerges in scenarios where data spans multiple orders of magnitude and is subject to certain constraints. One of the primary applications of Benford’s Law in internal audit is anomaly detection. When applied to financial data, it can uncover irregularities that may indicate errors, omissions, or even fraudulent activities.
The principle can be used to detect anomalies in several areas, including:
Revenue and Expenses: Consider a company’s revenue figures. According to Benford’s Law, the leading digit 1 should appear about 30 percent of the time. If a financial statement’s revenue figures significantly deviate from this expectation, it could warrant further investigation. Similarly, expenses, though often more variable, should still exhibit a discernible pattern according to Benford’s Law.
Accounts Payable and Receivable: In examining accounts payable and accounts receivable, Benford’s Law can provide a powerful tool for auditors. Any discrepancies in the leading digits of these figures might indicate errors or potential irregularities.
Inventory and Stock Data: Benford’s Law can be extended to various non-financial datasets. For example, in inventory management, the leading digits of stock levels can be analyzed. Deviations from the expected distribution might suggest discrepancies in recording or inventory shrinkage.
Tax Auditing: Benford’s Law has been employed in tax auditing as well. When applied to tax return data, it can help identify potential instances of tax fraud or manipulation.
Implementing Benford’s Law in Internal Auditing
Before applying Benford’s Law, it’s crucial to ensure that the data is well-structured and free from anomalies that could skew the analysis. Outliers and erroneous entries should be identified and corrected.
To rigorously apply Benford’s Law, statistical tests can be employed. The two-sample Kolmogorov-Smirnov test and the chi-square goodness-of-fit test are common methods used to assess the conformity of data to the expected distribution.
Rather than being a one-time tool, Benford’s Law can be integrated into ongoing auditing processes. Continuous monitoring of financial and non-financial data using this principle can help maintain a vigilant stance against potential irregularities.
While Benford’s Law is a powerful tool, it is not foolproof. Certain datasets, particularly those artificially generated, may not conform to this distribution. Additionally, the law does not provide insight into the nature or cause of anomalies; it merely signals their presence.
Benford’s Law stands as a testament to the remarkable patterns that underlie seemingly random data. In the realm of internal auditing, it offers a valuable tool for identifying anomalies and potential irregularities in financial and non-financial datasets. By leveraging this mathematical principle, auditors can enhance their ability to safeguard the integrity and accuracy of financial reporting, ultimately contributing to stronger, more resilient organizations. 
Jason Kincaid is a freelance business writer based in Fresno, California, who writes on the topics of data analytics and data governance.
Good one!, thanks for sharing. Rightly mentioned, it’s a symptom. The struggle I have noted especially when purchases follow a schedule of equated amounts over a period of time. In those cars the patterns tend to give false positives.