The intuition of Benford

Have you heard of Dr. Benford? What’s that you say? You haven’t? Well don’t worry you are not alone. Most people haven’t heard of the guy either, but it would be my advice to most anyone (especially business owners) to Google him.

You see back in the 1930s, Frank Benford was working away as a physicist. And being a physicist, he had to use a lot of math. And for the simple math, he would look up the answers in what is known as a book of log tables. (Now days we would just use a calculator but obviously it had yet to be invented)

For example, this could be how the book was laid out.

Pages 001-100 dealt with math problems starting with the number #1
Pages 101-200 dealt with math problems starting with the number #2
Pages 201-300 dealt with math problems starting with the number #3
Pages 301-400 dealt with math problems starting with the number #4
Pages 401-500 dealt with math problems starting with the number #5
Pages 501-600 dealt with math problems starting with the number #6
Pages 601-700 dealt with math problems starting with the number #7
Pages 701-800 dealt with math problems starting with the number #8
Pages 801-900 dealt with math problems starting with the number #9

Anyway, after several years of work, he started to notice something. Benford noticed that he used the pages at the front of the book a lot more than the ones at the back. Which means that he was doing the majority of his work with numbers that began with either 1,2, or 3. And this troubled him.

Am I a bad physicist? He thought. Am I only using a subset of the tools that are available?

And after several years of statistical work, Benford came to the conclusion that numbers seemed to form a pattern in most (but not all) logarithmic systems.

Numbers that began with #1 were used 30.1% of the time
Numbers that began with #2 were used 17.6% of the time
Numbers that began with #3 were used 12.5% of the time
Numbers that began with #4 were used 9.7% of the time
Numbers that began with #5 were used 7.8% of the time
Numbers that began with #6 were used 6.7% of the time
Numbers that began with #7 were used 5.8% of the time
Numbers that began with #8 were used 5.1% of the time
Numbers that began with #9 were used 4.6% of the time

Now, I know what you are thinking.

Jared, what the hell does this have to do with me?

And so here it is, in two words. Fraud detection.

That’s right, you can apply Benford’s law to most anyone’s financial books, and it will help detect if fraud is taking place.

Really?

Yes really! Take this little nugget from Wikipedia.

In 1972, Hal Varian suggested that the law could be used to detect possible fraud in lists of socio-economic data submitted in support of public planning decisions. Based on the plausible assumption that people who make up figures tend to distribute their digits fairly uniformly, a simple comparison of first-digit frequency distribution from the data with the expected distribution according to Benford’s law ought to show up any anomalous results.[5] Following this idea, Mark Nigrini showed that Benford’s law could be used as an indicator of accounting and expenses fraud.[6]

In the United States, evidence based on Benford’s law is legally admissible in criminal cases at the federal, state, and local levels.[7]

Benford’s law has been invoked as evidence of fraud in the 2009 Iranian elections.[8]

Pretty cool right? And please keep in mind, this is just one of many tools available to detect fraud. But for someone like myself who is responsible for large Oracle databases that process millions and millions of financial transactions a year, it is really nice to be able to use Benford’s law. It won’t actually solve any issues, but it will help raise those needed RED FLAGS. Not to mention the nerd in me had a lot of fun writing the algorithm :-)

So, to all you business owners who worry that someone is screwing with your money, it may be helpful to apply Benford’s law to your books, and see what pops up.

For the record, this it not legal advice, plus my field is computer science so this is more of a nerd out topic for me.

And here is a book with interesting information that I enjoyed reading (link)

Radiolab recently did a story on Math featuring Benford’s law (link)