Benford or Not Benford: A Systematic But Not Always Well-Founded Use of an Elegant Law in Experimental Fields


In this paper, we will propose a way to accurately model certain naturally occurring collections of data. Through this proposed model, the proportion of d as leading digit, \(d\in \llbracket 1,9\rrbracket \), in data is more likely to follow a law whose probability distribution is determined by a specific upper bound, rather than Benford’s Law, as one might have expected. These probability distributions fluctuate nevertheless around Benford’s values. These peculiar fluctuations have often been observed in the literature in such data sets (where the physical, biological or economical quantities considered are upper bounded). Knowing beforehand the value of this upper bound enables to find, through the developed model, a better adjusted law than Benford’s one.

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Correspondence to Stéphane Blondeau Da Silva.

Appendix: Python Script

Appendix: Python Script

Using Propositions 3.1, we can determine the terms of \((P_{(d,n)})_{n\in {\mathbb {N}}^*}\), for \(d\in \llbracket 1,9\rrbracket \). To this end, we have created a script with the Python programming language (Python Software Foundation, Python Language Reference, version 3.4. available at, see [24]). The implemented function expvalProp has two parameters: the rank n of the wanted term of the sequence and the value ld of the considered leading digit. Here is the used algorithm:


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Blondeau Da Silva, S. Benford or Not Benford: A Systematic But Not Always Well-Founded Use of an Elegant Law in Experimental Fields. Commun. Math. Stat. 8, 167–201 (2020).

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  • Benford’s Law
  • Leading digit
  • Experimental data

Mathematics Subject Classification

  • 60E05