# Benford’s Law: Fraud Detection by the Numbers

For most HTM professionals, accountancy is not your primary profession. Yet between contracts, capital equipment purchases, and parts spending, many HTM managers are, in effect, running multi-million dollar corporations. With such high dollar items like imaging glassware and ultrasound probes, the potential for fraud runs high. A less than honest employee looking to supplement his or her income can take advantage of chaos often found in a busy shop to help themselves to some high dollar items.

How can you guard yourself against wolves hidden amongst your flock?

The answer may lie in an interesting statistical phenomena known as Benford’s Law, also called the First-Digit Law. Benford’s law has been used by CPAs and fraud investigators to detect abnormalities since the 1930’s.

How does it work?

Imagine I have a bag containing 9 ping pong balls labeled 1 – 9. I ask you to close your eyes, reach into the bag, and pull out a ball. What is the probability you will pull out a 5? Probability theory tells us the answer is 1/9 or approximately 11%. That is because 1 ball out of the 9 in the bag is labeled 5. The same thing applies to any number you choose.

So now we are going to replace the bag of balls with a spreadsheet containing parts purchases over the past year. Looking at the column with purchase prices, I am going to take the first non-zero digit from each row. So if the price is \$130.00, I will take 1. If the price is \$67.34, I will take 6. Now, I will throw all of those first digits into my imaginary bag, have you reach in blindfolded and pull out a 5.

What do you think the probability was you would pull out a five? 11%? Actually it is 7.8%.

Huh?

I know what you are thinking. I have a bag of random numbers 1-9. Shouldn’t the probability of pulling any digit be 1/9 (11%)? Well not according to Benford’s Law.

Benford’s Law states that first digits taken from an organic set of numbers (i.e. numbers without artificial constraints) will follow the unique distribution seen below.

So the probability of pulling a 1 is 30% while the probability of a 9 is under 5%.

The distribution is calculated using the following formula:

Where the probability (p) of the number in question (n) can be calculated using a logarithm.

While it is a mathematical law, Benford’s Law is a phenomena meaning it is not fully understood how or why it works. The simplest explanation I can offer you involves looking at the following table demonstrating a 10% yearly compounded interest on a \$1 investment.

Notice how the first digit stays 1 for 8 years, but only 4 years at 2, and 1 year by the time it reaches 9. If you were to continue through 10’s to 100’s you would notice the same pattern repeating itself.

How can I use this?

Simply enough, take your parts purchasing history, truncate the first non-zero digit and create a histogram of your results. If it does not look like the distribution seen above, that is a good sign something is wrong. People trying to perpetrate fraud will often try to cover their tracks. Human intervention – altering order histories, removing purchases from ledger, will alter the organic nature of your data, and in turn, alter the distribution of your first digits.

While not the perfect catch-all by any means, Benford’s Law is used regularly by CPAs to alert them to possible fraud, book keeping errors, or other accounting irregularities. It is a simple tool which costs nothing to implement and should be part of the toolbox of any manager overseeing high volume spending.

## One thought on “Benford’s Law: Fraud Detection by the Numbers”

1. xylemuser

I like your article, very much indeed. I have also read somewhere that one digit analysis does not help much; however, two, three or even four digit analysis yields better results.

The trick is to develop the template, now that takes some time but when you do, the accountant or even auditors will be on their way of simplifying their work.