Benford’s Law – Using Maths to Catch Fraudsters

Benford’s Law is a very powerful and counter-intuitive mathematical rule which determines the distribution of leading digits (ie the first digit in any number).  You would probably expect that distribution would be equal – that a number 9 occurs as often as a number 1.  But this, whilst intuitive, is false for a large number of datasets.   Accountants looking for fraudulant activity and investigators looking for falsified data use Benford’s Law to catch criminals.

The probability function for Benford’s Law is:

benford 5


This clearly shows that a 1 is by far the most likely number to occur – and that you have nearly a 60% chance of the leading digit being 3,2 or 1.   Any criminal trying to make up data who didn’t know this law would be easily caught out.

Scenario for students 1:

You are a corrupt bank manager who is secretly writing cheques to your own account.  You can write any cheques for any amount – but you want it to appear natural so as not to arouse suspicion.  Write yourself 20 cheque amounts.  Try not to get caught!

Look at the following fraudualent cheques that were written by an Arizona manager – can you see why he was caught?   


Scenario for students 2:

Use the formula for the probability density function to find the probability of the respective leading digits.  Look at the leading digits for the first 50 Fibonacci numbers.  Does the law hold?

benford 4

There is also an excellent Numberphile video on Benford’s Law.  Wikipedia has a lot more on the topic, as have the Journal of Accountancy.

If you enjoyed this topic you might also like:

Amanda Knox and Bad Maths in Courts – some other examples of mathematics and the criminal justice system.

Cesaro Summation: Does 1 – 1 + 1 – 1 … = 1/2? – another surprising mathematical result.