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Latticework of Mental Models: Benford’s Law

On 25 November 2003, Kevin Lawrence was sentenced to 20 years in prison for pulling off possibly the biggest financial fraud in Washington State’s history. Here’s the backstory.

Kevin Lawrence graduated from high school in 1984. After a brief stint with a brokerage firm Lawrence bought a bowling alley and converted it into a fitness gym. He equipped the gym with modern exercise equipment, computers and hired chiropractors, masseuses and a nutritionist for the facility. But that was just the beginning of his entrepreneurship dreams. Soon he started working on an ambitious business plan to create a chain of high tech health clubs. He pitched the idea to a lot of investors.

Lawrence claimed that his startup would be an industry innovator that integrated fitness and health care into one business model, i.e., consumers could do fitness workouts and obtain health care within the same facility. His proposition also included offerings for design, manufacturing, and marketing of fitness equipments. Plus, he planned to build software to analyze the club member’s physical performance.

Lawrence must have been a good storyteller for he was able to convince more than two thousand investors and raise close to $100 million.

Flush with investor money, Lawrence floated two companies – Znetix Inc and Health Maintenance Centers Inc. Next, he started promoting Znetix through several marketing schemes such as sponsorship of athletes, teams, and events. Znetix splurged money on buying sports teams and lavish parties. These activities created an illusion in the investors’ minds that Znetix had backing from athlete celebrities and there was a viable fitness and healthcare business model.

In reality, there was no evidence that Znetix/HMC could make the business operation pay for itself. The company never filed its taxes and there were no financial controls in place. (Source: Kevin Lawrence Investigation)

Was it a case of an hyper-ambitious man being delusional and over-optimistic about a bad business? If Lawrence spent money only on promoting his business, it wouldn’t have raised suspicion. But Kevin and his cohorts were spending more money on personal items than on business.

Lawrence bought several properties including a home in Hawaii. He owned twenty personal watercrafts (including a 22-foot Bombardier speedboat), forty-seven luxury cars (five Hummers, four Ferraris, two DeThomaso Panteras, three Dodge Vipers, two Cadillac Escalades, a Lamborghini Diablo), Rolex watches, expensive diamond jewelry for his girlfriend(s) and a $200,000 Samurai sword.

I read about Kevin’s story in Leonard Mlodinow’s book The Drunkard’s Walk. Mlodinow writes –

Lawrence and his pals tried to cover their tracks by moving investors’ money through a complex web of bank accounts and shell companies to give the appearance of a bustling and growing business. Unfortunately for them, a suspicious forensic accountant named Darrell Dorrell compiled a list of over 70,000 numbers representing their various checks and wire transfers and compared the distribution of digits with Benford’s law. That, of course, was only the beginning of the investigation, but for there the saga unfolded predictably, ending the day before Thanksgiving 2003, when, flanked by his attorneys and clad in light blue prison garb, Kevin Lawrence was sentenced to twenty years without possibility of parole.

Znetix investigation went on for three years during which the Department of Financial Institutions spent an estimated 12,000 hours. The investigation team traced thousands of bank transactions. They pored over nearly 570 bank accounts and about 600,000 cheques, deposit items and wire transfers. That sounds like like a task more tedious than a plane crash investigation. However, most of those 12,000 hours were about finding confirming evidence that there was a fraud.

The real credit goes to Darrell Dorell for recognizing the anomaly. In fact, not even Darrell, it was Benford’s law which gave away Lawrence’s secret. So what is Benford’s Law?

In my 13-year career working with half a dozen different software companies, I have gone through numerous personality assessments and psychometric tests. I understand that my employers were being genuinely concerned about my career and wanted me to take those quizzes seriously. But, honestly, answering those 100 multiple choice questions, especially when I was right in the middle of an intense struggle to meet my project deadline, was more of a nuisance than a help.

So what did I do? I finished the test in 5 minutes by ticking option A for every question.

“Oh! C’mon. That’s too obvious. Anyone looking at your answers would’ve figured out that you were not answering honestly.” You might want to argue.

I know. That’s why I ticked the answers randomly. So a pattern like A-A-B-A-C-B-C-B-B-A-C won’t raise any suspicion, right? Unless my truly random pattern turned out to be the exact sequence preferred by a sociopath killer. Thankfully, it wasn’t because I was never sent to jail based on those psychometric tests.

The point I am coming to is this: we humans have no intuitive understanding of randomness. Humans are easily fooled by randomness. Taleb in his book Fooled by Randomness argued that our world is more random than we think.

Benford’s law says, when it comes to large numerical datasets, our world is less random than we think.

Benford’s law is an observation about the frequency distribution of leading digits in many real-life sets of numerical data. The law states that in many naturally occurring collections of numbers, the leading significant digit is likely to be small. For example, in sets that obey the law, the number 1 appears as the most significant digit about 30% of the time, while 9 appears as the most significant digit less than 5% of the time. By contrast, if the digits were distributed uniformly, they would each occur about 11.1% of the time. (Source: Wikipedia)

Image Source: Wikipedia

So what do we mean by real-life sets of numerical data?

If I were to collect the age-data of all the people in a town or a country, will the age distribution depict Benfords’ law?

No. It won’t because the range of data in such case would be very small, i.e., from 0-122 (the oldest known person was 122 years old at the time of her death). Wikipedia states:

Benford’s law tends to apply most accurately to data that are distributed uniformly across several orders of magnitude. As a rule of thumb, the more orders of magnitude that the data evenly covers, the more accurately Benford’s law applies.

Which means the prime candidate where Benford’s law would be observed very frequently, as you’d have guessed by now, is financial data. Someone “cooking the books” will likely use an even distribution of leading digits rather than leading with low digits. Like I tried to fudge my psychometric tests. Fortunately, Benfords’ Law doesn’t work with multiple choice personality assessment test data.

Darrell Dorrell wasn’t first to use Benford’s law for detecting a financial fraud. The law has been in use since 1972 for unearthing accounting frauds.

Mark Nigrini, the author of Forensic Analysis, has shown that it could be used in forensic accounting and auditing as an indicator of accounting and expenses fraud.

Here are few more interesting instances where Benford’s law has been used:

  • Benford’s Law was invoked as evidence of fraud in the 2009 Iranian elections.
  • The macroeconomic data the Greek government reported to the European Union before entering the eurozone was shown to be probably fraudulent using Benford’s law.
  • One researcher even applied the law to thirteen years of Bill Clinton’s tax returns. He passed.

Stephanie, in her blog Statistics How To, writes about her experiment with Benford’s law. She picked up the Oct 2016 issue of TIME magazine and counted all the numbers appearing in it. Surprisingly, it adhered to Benford’s law. Of course, that’s anecdotal evidence and doesn’t prove anything. However, that gave me an idea to do the same experiment with a company’s annual report.

So I randomly picked up an annual report. I don’t deny the role of my subconscious in this random selection. I wrote a small program to extract all the numbers from this annual report and plotted the frequency distribution. This is what I got –

Compare it with the first image that shows Benford’s frequency distribution. What do you see?

Now before you jump to any conclusions, I would like to put out a disclaimer.

This was a quick and sloppy experiment. I spent less than 30 minutes. Ideally, I should have scrubbed the numerical data before running my analysis, i.e., I should have filtered out the numbers which were not really financial data like the page numbers, section numbers, etc. I didn’t do all that.

So please don’t rely on these results. It was an experiment more for fun rather than investigation. By the way, it was Reliance Communication’s FY17 annual report.

Thanks for reading.

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About the Author

Anshul Khare worked for 12+ years as a Software Architect. He is an avid learner and enjoys reading about human behaviour and multidisciplinary thinking. You can connect with Anshul on Twitter.


  1. Sourabh Jain says:

    Thanks for this interesting article. I was wondering how this has relevance or can be used by a retail investor for making investments in stocks?

  2. What software did you use to extract numbers from the annual report?

    • Anshul Khare says:

      First, I converted pdf file to text file using an online converter (Google search will help you find many such online services)
      Second, wrote a small Java code to parse the text file and filter out the non-digit words.

  3. Hi, Please correct me if I am wrong,
    Mark Nigrini, who found fabricated tax returns fraud states that “True tax data usually follows Benford’s law, whereas made-up returns do not”
    So if annual report is showing numbers as per Benford’s law, that means as per theory, there are no “chances of manipulation” with the numbers (I am not saying the annual report has or do not have any issues)

    • Anshul Khare says:

      Sherlock Holmes said, “absence of evidence is not evidence of absence.”

      As per my understanding, if the numbers follow Benford’s law distribution then it doesn’t really tell us anything. Maybe the guy who made up the number knows about Benford’s law so he doctored the numbers accordingly.

      If numbers don’t follow Benford’s law then there a possibility that those numbers have been cooked and you need to dig deeper before concluding anything.

  4. Benford law will fail on the Bata shoes pricing. Bata prices its shoes as per following series –
    Rs. 99.99
    Rs. 499.99
    Rs 899.99
    Rs. 999.99
    Rs 2999.99
    Rs 3999.99
    So you wont find the leading significant digit to be small here as stated by Benford law.
    Your crude software will also fail if we feed Bata shoe prices data to it.

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