"Looking out for number one": Benford’s Law for procurement & compliance officers - to root out fraud, bribery & corruption ^Featured

"Looking out for number one"

Go and look up some numbers - any naturally occurring numbers will do, such as the length of rivers, population sizes etc. Then, taking these numbers and ignoring any leading zeroes, record how many begin with 1, 2, 3 and so on

Intuitively, you might think that there would be roughly the same number of numbers that begin with each different digit i.e. the proportion of numbers beginning with each different digit would be roughly 1/9^th or 11.1%. However, in a lot of cases you would be wrong and, surprisingly, for many types of numerical data, the distribution of first digits is highly skewed, with 1 being the most common digit and 9 the least common

There is a precise mathematical relationship behind this phenomenon - a logarithmic relationship - which is known as Benford’s Law, or the first-digit law

Benford’s Law

Benford's Law states that in lists of numbers from many real-life sources of data the leading digit is distributed in a specific and non-uniform way. According to this law, the first digit is 1 about 30% of the time, 2 about 18% and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than 5% of the time

This counter-intuitive result has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates and lengths of rivers to name but a few

This digit distribution phenomenon was first identified in 1881 by Simon Newcomb, a mathematician and astronomer, when he noticed that the pages of a book of logarithms became more worn the closer one was to the front, implying that the lower numbers were used more frequently. Frank Benford, a physicist, noticed the same pattern in 1938 and, after collecting an enormous amount of data, concluded that about 30% began with 1, 18% with 2 and so on. His analysis was evidence for the existence of the law, but Benford, also, was unable to explain quite why this should be so, which eventually took until the 1990’s

Using Benford’s Law

Whilst Benford's Law gives an interesting and surprising result, it has direct relevance to the work of procurement and compliance professionals in the areas of fraud, bribery and corruption

One good example is that if someone attempts to falsify an accounting return then, invariably, they will have to invent some data. When trying to do this, the tendency is for people to use too many numbers starting with digits in the mid range such as 5, 6 or 7 - and not enough numbers starting with 1. Clearly, this violation of Benford's Law would immediately set alarm bells ringing

As another example, if Benford’s Law had been applied prior to the Enron scandal then it would have revealed issues with the earnings reports released in 2001 and 2002; revenue numbers were subject to upwards management, as were the Earnings per Share (EPS) numbers which showed a marked discontinuity in the distribution

In procurement and compliance, Benford’s Law can be used to show any skew in invoice values; typically, this analysis will highlight that invoices are concentrated around, say, 4 - to get below a signing threshold of £500 or £5000. If suppliers are attempting to circumvent such rules, usually with the collusion - tacit or otherwise - of accounts payable, then what else is going on?

Benford and the lottery

Sadly, Benford’s Law will not help with the lottery. The outcome of the lottery is genuinely random, such that every possible lottery number has an equal chance of appearing. The leading-digit frequencies should therefore, in the long run, be in exact proportion to the number of lottery numbers starting with that digit

On the other hand, consider times for the marathon and other data sets where few, if any, of the numbers begin with 1. Unlike the lottery, this data is not random; instead, it is constrained and too narrow for a law of digit frequencies to work

In other words, Benford's Law needs data that is neither totally random nor constrained, but instead lies somewhere in between. However, this is exactly the type of data that is encountered in real-life business situations and that is why Benford’s Law is such a useful - and simple - tool

Links & Sources - with thanks

• +plus magazine                      http://plus.maths.org/content/looking-out-number-one
• Wikipedia                               http://en.wikipedia.org/wiki/Benford's_law
• Wolfram Math World               http://mathworld.wolfram.com/BenfordsLaw.html
• Rex Swain                              http://www.rexswain.com/benford.html