# Mike-amatical’s Fun Fact #1

Guest Writer: Mike Streinz HMC ’20

What do financial fraud, an electrical engineer, and the American rock band Three Dogs Night have in common? The number one! In the case of Three Dogs Night one of their top hits in 1968 is titled “One is the Loneliest Number.” However, three decades earlier the contrary was discovered by an electrical engineer named Frank Benford. Benford tallied the first digit of numbers collected from US population statistics, street addresses of colleagues, and baseball statistics and found a fascinating quirk of data! The leading digit of the numbers that he observed was the number one more than 30% of the time, far greater than a uniform distribution. On top of that, he found that as a leading digit increased, its frequency as a leading digit decreased. So perhaps one isn’t the loneliest number.

Benford’s law was formally discovered in 1938, but the phenomenon was first noticed nearly half a century earlier. Astronomer Simon Newcomb published a brief note explaining the concept after he noticed books of log tables were significantly more worn down towards the beginning of these books. Since values were listed sequentially, values with a smaller leading digit appeared first. From this observation he made the conjecture that the probability a number began with some digit d, was log(d+1)-log(d). The implication of this is the number one appears as a leading digit 30% of the time, as observed by Benford, and nine a paltry 5%.

The proof of Benford’s Law is both long and complicated so we’ll settle for the simplest explanation. If you picked a random number from one to one thousand and then counted up to that number, it’s most likely that one was the most frequent leading digit of all the numbers that you counted. Simply put, you have to count numbers with a leading one first. You have to count the tens before the twenties and the thousands before two, three, or four thousand. Unfortunately, this puts some limits on where Benford’s law can apply. Benford’s law only occurs in data that covers multiple orders of magnitude. In addition, the data sets must be naturally random and large.

So how does financial fraud relate to all this mess? Benford’s Law has been used recently to identify cases of financial fraud and is considered admissible evidence in federal, state, and local court. An example of its use was an instance when one fraudster wrote numerous checks to himself for values just under \$100,000. Values over that threshold got audited, however, because the checks were near the threshold, and the leading digit was often nine. Benford’s Law did not hold and that alone was enough to alert authorities. Present day discoveries haven’t stopped. Applications to various fields are currently being explored, elevating the importance of Benford’s Law daily.