Theranos, Elizabeth Holmes and Benford’s Law. Most people have never heard of Benford’s Law but are well aware of the massive corporate fraud at Theranos and Ms. Holmes (see the movie Bad Blood). The question is – could Benford’s Law have been used to detect the Theranos fabricated blood test results early on and saved investors billions?
In 2015, Forbes estimated Elizabeth Holmes’s net worth to be $4.5 billion thanks to the company she founded at just 19 years old. Theranos was touted as a revolutionary blood-testing company that could produce results with just a pinprick of the finger. It is estimated her investors lost nearly $1 billion. Today, her net worth is estimated at zero or less.
Astronomer Simon Newcomb, in 1881, noticed that the page numbers in a book were worn more toward the front of the book and progressively less worn toward the end of the book. Newcomb recognized a distinct pattern to the occurrence of lower numbers versus higher numbers. He published a scholarly article explaining his observations surrounding the probability of a single number being the first number of a longer number.
Fifty-seven years later Frank Benford tested Newcomb’s hypothesis and garnered much of the credit for developing what is now known as Benford’s Law. Benford attempted to explain it by saying “it is easier to own one acre than nine acres,” implying that more people own one acre than own nine acres. Thus, in a population of landowners, the frequency of the numbers beginning with 1, 10, 100 and so on would be predictable. The following chart illustrates the normal distribution of the leading digit in a sample of numbers:
In 2017, J. Carlton Collins published an article simplifying how this works. “Using a nontechnical description, Benford’s Law works whether you are counting dollars, acres, inventory, populations, or anything, because you must first count 1 before counting 2, 3, or 4; you must first count 10 before counting 20, 30, or 40; you must count 100 before counting 200, 300, or 400; and so on.
Every counting job starts with lower numbers, but not all counting jobs progress to include the increasingly higher numbers. As an example, when counting to 25, 11 numbers have a leading digit of 1, seven numbers have the leading digit of 2, and only one number leads with the digit 3. Even in this simplified counting exercise, you can see Benford’s Law at work—rather than each digit having an equal chance at being the first digit, lower numerals always have a greater chance at being the leading digit compared to higher numerals.”
Applying Benford’s Law by Tommie W. Singleton, PhD. ISACA Journal Archives 2011 – “For instance, if a bank’s policy is to refer loans at or above US $50,000 to a loan committee, looking just below that approval threshold gives a loan officer the potential to discover loan frauds. If loan fraud was being perpetrated, a Benford’s Law test of looking at either the leading digit (specifically, the 4) or two leading digits (specifically, 49) has the potential to uncover the fraud.
Figure 2 shows what a Benford’s Law test of the leading digit might show as a result in this particular scenario. The line is Benford’s Law probabilities and the bars are the actual occurrences. Note that 4 is aberrantly high in occurrence, and 5 is too low, indicating the possible manipulation of the natural occurrence of loans beginning with 5 (US $50,000 loans) possibly being switched to just under the cutoff or indicating that the suspect could be issuing a lot of $49,999.99 loans fictitiously to embezzle funds.”
Is Benford’s Law admissible in Court? The answer is yes. If applied correctly, it should be admissible under Daubert and Rule 702. See United States v. Channon (Matthew), No. 16-2254 (10th Cir. 2018).
If you are so inclined, you can convince yourself the law works. Pick up a random book or magazine and sort the numbers – about 30% of the numbers collected will start with the number 1. The result is always the same: 30% start with the number 1. Using this fact, a fraud auditor has a simple tool to examine large sets of numbers for irregular results.
The answer to the above question, “could Benford’s Law have been used to detect the Theranos fabricated blood test results early on and saved investors billions,” is yes. A fraud investigator could have used Benford’s Law to detect the occurrence of fraudulent lab results at Theranos. The Law is not proof of fraud, but it will highlight irregularities for further investigation.
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