Benford’s law is quirky, fascinating and useful in detecting fraud or even errors and anomalies. I love it. The quirkiness starts with the name, because it is not actually a law. It is a predictor of digits. Ignoring zeros for the moment (because no number starts with zero), every number has one or more of the digits 1 to 9. Most people (including me) would think that any one of the digits 1 to 9 would have an equal chance of being the first digit in any given number. However, in most real-life situations, this is not so. In fact, the digit 1 appears first in about 30% of numbers. Digit 2 is first in about 18% of numbers and so on downwards to digit 9 which appears first in about 5% of numbers. For example, in the number 1863, digit 1 is the first digit written down. To put it another way, Benford’s law predicts what the left-hand number might be. (Actually, it also predicts what the second, third and fourth numbers might be, but let’s not complicate).
Benford’s Law works in hundreds of real-life sets of numbers. It might be populations of countries, stock market values, house prices or sports statistics. The list is long. Try it with all the numbers in the newspaper you are reading. (It works best when there are lots of numbers).
This turns out to be extremely useful for accountants and auditors. It can indicate the possibility of fraud. Let’s say an employee is creating false invoices and directing the payments to his own bank account. He chooses random amounts to put on the invoices. Benford’s is going to trap him, because it is almost impossible for someone manipulating the system to mimic the pattern predicted by Benford’s law.
Anomalies do not necessarily have to be the result of fraud. They can be caused by error or weaknesses in the system. Either way, they are useful red flags.
Accountants and auditors can export numbers from a set of accounts to an Excel spreadsheet and match them with Benford’s predictions. The results are always fascinatingly reliable. After using Benford’s for a dozen years or more, it has never failed to predict the frequencies of the left-hand numbers. For me, it is like watching a big plane lift off the ground. I never fail to be amazed.
Nobody devised Benford’s law. Benford (and others) just happened to notice that this was so. There is a paper by RM Forster called “A Simple Explanation of Benford’s Law”. It is extremely complicated. I still do not know why it works, but it does and I love it.
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