Understanding Benford’s Law

Benford’s Law is a difficult one to accept, but I have personally spent many hours checking randomly different sets of data and there is not doubt Benford’s Law is a fact.

We all know the 26 letters of the Alphabet and can understand that different letters have different usage rates:

The top ten most common letters in the Concise Oxford English Dictionary, and the percentage of words they appear in, are:

  • E – 11.1607%
  • A – 8.4966%
  • R – 7.5809%
  • I – 7.5448%
  • O – 7.1635%
  • T – 6.9509%
  • N – 6.6544%
  • S – 5.7351%

But numbers? there are only 10 digits 0,1,2,3,4,5,6,7,8,9 you would think the usage of numbers would be spread evenly across all 10, the laws of probability tell us so, however, it is not the case.

Numbers are used in a very clear pattern.

Benford's law - Wikipedia

That is correct, in naturally occurring data the number 1 appears a whopping 30% of the time and 1,2,3 make up over 50% of all such data.

A number of criteria – applicable particularly to accounting data – have been suggested where Benford’s law can be expected to apply and not to apply. Distributions that can be expected to obey Benford’s law

  • When the mean is greater than the median and the skew is positive
  • Numbers that result from mathematical combination of numbers: e.g. quantity × price
  • Transaction level data: e.g. disbursements, sales

Distributions that would not be expected to obey Benford’s law

  • Where numbers are assigned sequentially: e.g. check numbers, invoice numbers
  • Where numbers are influenced by human thought: e.g. prices set by psychological thresholds ($1.99)
  • Accounts with a large number of firm-specific numbers: e.g. accounts set up to record $100 refunds
  • Accounts with a built-in minimum or maximum

Fraud examiners use Benford’s Law tests on natural numbers, like payment amounts. The theory is that if a fraudster submits fake invoices for payment, he won’t submit invoices for $100 or $200, he will want to go big and submit invoices for $900 or $800. If you do that enough times, it upsets the natural order of the way numbers should occur (according to Benford). For example, if you run a Benford’s Law test on your April payments, and you find the first digit was 9 in 35% of the payments, that’s an anomaly. Bendford’s Law says 9 should be the first digit only 4.6% of the time.

No test is foolproof, but Benford’s Law does provide an extra method for fraud examiners to test data for potentially fraudulent activity. You can read more about Benford’s Law in the online Fraud Examiners Manual.

The following two tabs change content below.
S. Jack Heffernan Ph.D. Funds Manager at HEFFX holds a Ph.D. in Economics and brings with him over 25 years of trading experience in Asia and hands on experience in Venture Capital, he has been involved in several start ups that have seen market capitalization over $500m and 1 that reach a peak market cap of $15b. He has managed and overseen start ups in Mining, Shipping, Technology and Financial Services.

Latest posts by S. Jack Heffernan Ph.D (see all)