In 1931, Frank Benford stated a law about the frequency distribution of leading digits in numerical data sets which was somewhat contradictory to general common sense. His First-Digit law claims that for many real-life measurement data, such as heights of buildings, lengths of rivers, population numbers and death rates, the appearances of digits 1 to 9 are not weighted equally (i.e. 11.1% each) but rather disproportionately with more weight toward smaller digits (around 30% for 1 and only 5% for 9). The law also predicts the frequency distribution of the remaining digits, which approach a uniform distribution.

The actual discovery of the law dates back to 1881, when astronomer Simon Newcomb noticed that the earlier pages of logarithm tables (containing numbers starting with 1) were much more worn than the rest of pages. Since then, the validity of the law has been investigated via several real-life examples and it has been shown that the law stands true regardless of the unit used for measurement. However, for the law to apply best, the data must fall into a wide range of orders of magnitudes. The reason behind this is that the key to this incident is assuming a uniform distribution for the logarithm of the data instead of the data itself. Friar (2012) demonstrated Benford’s distribution for genomic sizes. Mir (2014) studied the Benford law behavior of the religious activity data.

Using the fact that many are unaware of this phenomenon, Benford’s law has also found application in detecting fraud in different areas from accounting to election to science. Nigrini (1996) studied the use of Benford’s law as an expected distribution for the digits in tabulated data to facilitate the detection of tax evasion. Durtschi (2004) promoted Benford’s law for developing a tool to assist the auditor in detecting fraud in accounting data more effectively. Moreover, Mebane (2006) studied the presidential election votes by testing whether the second digits of the reported vote counts obey the frequencies dictated by Benford’s law. Diekmann (2007) applied Benford’s law to detect fabricated or falsified scientific data and fraudulent financial data.

Although, for many years, it was thought that Benford’s law is solely an interesting mathematical observation, it’s achievements in recent years have made it a promising tool for more sophisticated future applications.

References:

https://en.wikipedia.org/wiki/Benford’s_law

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