Benford’s Law

Benford’s Law is names after the American physicist Frank Benford who described it in 1938, although it had been previously mentioned by Simon Newcomb in 1881.

Benford’s Law states that in “naturally occurring collections of numbers” the leading significant digit is more likely to be small. For example, in sets of numbers which obey this law, the number 1 appears as the first significant digit about 30% of the time, which is much greater than if the digits were distributed uniformly: 11.1%.

In mathematics, a set of numbers satisfies this law if the leading digit, d, occurs with a probability:

P(d)=\log _{10}(d+1)-\log _{10}(d)=\log _{10}\left({\frac {d+1}{d}}\right)=\log _{10}\left(1+{\frac {1}{d}}\right).

Hence, if d = 1, then P(1) = log 2 = 30.1..%

The leading digits have the following distribution in Benford’s law:

Screen Shot 2016-09-27 at 2.57.40 PM.png

Source: Wikipedia

BenfordsLaw

Source: Wolfram MathWorld

As P(d) is proportional to the space between d and d+1 on a logarithmic scale, the mantissae of the logarithms of the numbers are uniformly and randomly distributed.

Applications

Benford’s law has found applications in a big variety of data sets, such as stock prices, house prices, death rates and mathematical constants.

File:Benford-physical.svg

Frequency of first significant digit of physical constants plotted against Benford’s Law | Source: Wikipedia

Due to this, fraud can be found applying Benford’s law to data sets. This is because if a person is trying to fabricate ‘random’ values to try to not appear suspicious, they will probably select numbers such that the initial digits are uniformly distributed. However, as explained above, this is completely wrong! In fact, this application of Benford’s law is so powerful that there is an “industry specialising in forensic accounting and auditing which uses these phenomena to look for inconsistencies in data.”

Datagenetics.com describes how:

“In 1993, in State of Arizona v. Wayne James Nelson (CV92-18841), the accused was found guilty of trying to defraud the state of nearly $2 million, by diverting funds to a bogus vendor.

The perpetrator selected payments with the intention of making them appear random: None of the check amounts was duplicated, there were no round-numbers, and all the values included dollars and cents amounts. However, as Benford’s Law is esoterically counterintuitive, he did not realize that his seemingly random looking selections were far from random.”

Sources: 1 | 2 | 3 | 4

 

After writing this I’ve realised that I touched on this law (in less detail) in a previous post during my Christmas series! M x

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