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Benford’s Law and the mysteries beneath the numbers

Accounting is centered on numbers. Every day accounting journal entries full of numbers are recorded, accumulating as each day passes. By year-end, the general ledger will contain hundreds of thousands of entries, if not millions. As we roll up our sleeves to perform the year-end close, we are tasked with making sense of it all and, in most cases, solving number mysteries. Someone once said that numbers have life; they’re not just symbols on paper. Frank Benford shared a similar thought as he began investigating patterns in numbers that manifested a strange phenomenon.

Frank Benford, a General Electric physicist/electrical engineer born in 1883, noticed that the first pages of books were more worn than the later pages. Inspired, he began investigating over 20,000 numbers to analyze patterns surrounding the use of numbers in comparison to other numbers. What he concluded was the infamous Benford’s Law.

Benford’s Law simply states that smaller digits are more likely to lead a number than large digits. The higher the number becomes, the less likely it is to lead a number. For example, the numbers 1, 10, 11, 12, 13, 100, etc. will appear more often than 3, 33, 37 and 300, and those will appear more often than 9, 91, 92 and 901. In a large natural collection of numbers, the number 1 appears as the leading digit about 30% of the time, while 9 appears as the leading digit less than 5% of the time. Don’t believe it? We put it to the test by taking an annual check register from a mid-size company and ran it through Benford’s analysis. We ran the analysis over the first two digits of each number and this is the graph we retrieved:

The results reveal that there were 180 numbers beginning with 10 followed by a downward trend to numbers beginning with 99 only totaling 19. Without any manipulation, the set of data mystically fell in line with Benford’s theory. The only exceptions were the numbers 25 and 50, which occurred more than expected. After further review of the check register, we noted that the company made numerous \$250 and \$50 payments for recurring normal business expenses that were validated.

In the world of auditing, Benford’s Law can be a valuable tool. It is becoming more common for auditors to use data analytics and tools such as Benford’s Law to identify any instances of potential fraud. It is an easy way to analyze a large amount of data. Studies have supported the legitimacy of Benford’s distribution curve and have concluded that companies with financials statements that do not conform to Benford’s distribution have a greater chance of having poor internal controls and fraud. While Bedford’s Law should not be used as a decision-making tool by itself, it may prove to be a useful tool to indicate that a general ledger detail, check register or financial statement deserves a deeper analysis.

In conclusion, I hope this has opened your mind to the world of numbers and the mysteries that lie beneath them. In the words of Yoda, “Much to learn you still have my old padawan. This is just the beginning.”

Joe Bang, CPA