Benford's law, also called the first-digit law, was made famous in 1938 by Physicist Frank Benford, who after observing sets of naturally occurring numbers, discovered a surprising pattern in the occurrence frequency of the digits one through nine as the first number in a list. In essence, the law states that in numbered lists providing real-life data (e.g., a journal of cash disbursements and receipts, contract payments, or credit card charges), the leading digit is one almost 33 percent (i.e., one third) of the time. On the other hand, larger numbers occur as the leading digit with less frequency as they grow in magnitude to the point that nine is the first digit less than 5 percent of the time.
In the 1970s, Hal Varian, a professor at the University of California's Berkeley School of Information, suggested that the law could be used to detect possible fraud in lists providing socioeconomic information. Since then, Benford's law has been applied to large numbers of data to detect unusual patterns that are often the result of errors or, worse, fraud. As part of their work, internal auditors often employ tools and scientific methods that enable them to detect instances of fraud. Although the use of Benford's law might seem daunting at first, auditors don't need to have advanced degrees or an expensive data analysis tool to use Benford's law as part of their fraud investigations ― this task can be effectively and efficiently performed using Microsoft Excel.
A First Glance into Benford's Law
When getting to know Benford's law, auditors need to understand the true definition of naturally occurring numbers (i.e., those numbers that arise naturally from real-life sources). Population numbers, death rates, baseball statistics, chemical constants, or financial transitions are all examples of naturally occurring numbers. On the other hand, numbers that are largely controlled or assigned by people are not considered naturally occurring numbers. These include product serial numbers, customer account numbers, and zip codes.
Benford's law states that if there is a set of non-manipulated, naturally occurring numbers, the occurrence frequency of digits one through nine as the first digit should be expected. As we can see from the numbers in table 1, naturally 30 percent of numbers have one as a leading digit, and nine occurs as a leading digit only one time in twenty. Because most financial and accounting data conform to naturally occurring numbers, by comparing the occurrence frequency of these first digits to Benford's pattern, auditors should be able to determine irregularities and possible manipulations.
Table 1 Pattern of naturally occurring numbers |
For example, let's assume an employee is committing fraud by creating and sending payments to a fictitious vendor. Since the amounts of these fraudulent payments are made up rather than occurring naturally, the leading digit of all fictitious and valid transactions will no longer follow Benford's law. Furthermore, assume many of these fraudulent payments have three as the leading digit, such as $39, $322 or $3,187. By performing a first-digit test on the disbursement data using Benford's law, auditors should see the amounts that have the leading digit three occur more frequently than the usual occurrence pattern of 12.5 percent.
Although Benford's law has been proven in various situations to be effective in detecting data manipulations, it has its limitations. In some circumstances, a set of naturally occurring numbers may not always follow the law due to human interventions. If the numbers are limited to a specific range, for instance, such as a price range of $7.99 to $9.99 that is set by the vendor, this set range will cause the pattern to be shifted or limited to a particular area. In this case, the auditor has to decide whether Benford's law should be used as a testing method.
Applying Benford's Law
The attached PDF (PDF, 671 KB) contains step-by-step instructions on how to apply Benford's law using Microsoft Excel, which is commonly used by internal auditors around the world in their day-to-day work. The technique is explained in the context of a realistic example and should enable auditors to easily and effectively apply Benford's law to their company's data when identifying unusual data patterns that may signal the presence of errors or fraud.
Making the Most Out of Benford's Law
In our example, there is an unusual amount of transactions where the leading digit begins with three. This indicates that there are possible errors or manipulations in the transactions. Auditors should perform further tests, such as a two-digit test, on these transactions, especially on the ones that have the suspicious digit (i.e., the digit three in this example).
Most business data, such as sales numbers, expenses, accounts receivables, disbursements, and even the number of customer street addresses, can be considered naturally occurring numbers. By comparing the first-digit frequency distribution of naturally occurring data with Benford's expected distribution, auditors will be able to easily spot possible errors or fraudulent transactions. Therefore, when used correctly, Benford's law can be a useful and inexpensive tool for identifying suspect accounts for further analysis.
For more information on Benford's law, auditors can read:
Wikipedia's Benford's law entry
"Digital Analysis: A Computer-assisted Data Analysis Technology for Internal Auditors" published in ITAudit's Dec. 15, 1998 issue
"I've Got Your Number" published in the AICPA's Journal of Accountancy May 1999 issue