26/05/2021

Folha de S. Paulo: Viana writes about the magic of Benford's Law

Foto: Freepik

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

In 1881, astronomer Simon Newcomb discovered that in astronomical data, it is much more common for the initial digit (the leftmost one) to be small (1, 2, …) than large (9, 8, …). The chance of a number starting with the digit d is approximately equal to the decimal logarithm of (1+1/d): for d=1 this gives 30.1%, but for d=9 it is only 4.6%.

This surprising fact was rediscovered in 1937 by physicist Frank Benford, who pointed out that the same thing happens with many different types of data: city populations, numbers in this edition of Folha de S.Paulo newspaper, Covid cases in different countries and states, volcano sizes, time intervals between heartbeats, points in basketball games, etc.

There are exceptions, such as when the data is artificial (cell phone numbers in Rio de Janeiro always start with 9) or varies within a limited range (adult heights in feet almost always start with 5 or 6). But it is widely confirmed that the vast majority of natural data follows Benford's Law, as I illustrated in the last edition of this column. This means that it can be used to help distinguish real data from incorrect or fraudulent data.

One application is in auditing tax returns: if the return is genuine, the values should comply with Benford's Law; therefore, any discrepancy is an indication that the return should be flagged for further review and carefully analyzed. The Brazilian Federal Revenue Service and its counterparts in other countries do not reveal their methods, so we are unaware of how widely used the law actually is. But it is a free tool and very easy to apply.

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