02/06/2021

Marcelo Viana in Folha: What is Benford's Law for?

Crédito: Freepix

Reproduction of Marcelo Viana's column in Folha de S.Paulo

A reader kindly sent me a 2011 study in which Benford's Law, applied to the economic data of European Union countries, raised suspicions of manipulation of Greek data, which was later officially confirmed.

The law, discovered by S. Newcomb in 1881 and F. Benford in 1937, states that in most types of data, the chance that the initial digit (the leftmost one) is d is given by the decimal logarithm of (1+1/d). This chance decreases as the digit increases: for d=1 it is 30.1%, but for d=9 it is only 4.6%.

We don't know exactly why this law works, but it's a useful tool for detecting all kinds of fraud, not just accounting fraud. It's used, for example, to audit data presented in scientific articles: in genuine articles, the information follows Benford's Law, but not in "fabricated" works.

Adherence to the law must apply regardless of the unit used (inch, meter, or light-year, for example), although the numbers change significantly depending on the unit. This means that even if the "manufacturing" is perfect in the unit used in the article, it can still be identified simply by changing the unit.

Another property that makes it difficult to deceive Benford's law is that it holds true in any number base: if we replace 10 with another base, b, the chance that the initial "digit" is d is given by the logarithm of (1+1/d) in that base b.

To read the full text, visit the newspaper's website.