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An Alternative to the Oversimplifying Benford’s Law in Experimental Fields


A way to model the distribution of first digits in some naturally occurring collections of data is here highlighted. The proportion of d as leading digit, d1,9, in data is sometimes more likely to follow a specific law whose probability distribution is determined by a lower and an upper bound, rather than Benford’s Law, as one might have expected. These peculiar probability distributions fluctuate around Benford’s values, such fluctuations having often been observed in the literature in experimental data sets (where the physical, biological or economical quantities considered are lower and upper bounded). Knowing beforehand the values of these bounds enables to find, through the developed model, a better adjusted law than Benford’s one.

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Appendix A: Python Script

Appendix A: Python Script

Using Proposition 3.1, we can determine the terms of \(({P_{N}^{n}}(d))\), for d1,9. 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 Van Rossum (1995)). The implemented function Prob has three parameters: the value of n, 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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Silva, S.B.D. An Alternative to the Oversimplifying Benford’s Law in Experimental Fields. Sankhya B 84, 778–808 (2022).

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