The importance of being “one” (or Benford’s law)

Abstract

In this article we give an informal presentation of the so-called Benford’s law, a counterintuitive probability distribution that describes the frequency with which the first significant digits—that is, a digit between 1 and 9—appear in sets of numbers associated with heterogeneous quantities. Unlike what might be expected, these frequencies are not equal but are decreasing, that is, 1 appears more often than 2, which appears more often than 3 and so on. After an introduction to the law, we will show how, perhaps unexpectedly, it is satisfied by the numbers generated by some sequences of powers. We will then illustrate some applications in the socio-economic field (for instance, to identify accounting or electoral fraud). Finally we will use it to analyse the tax returns of some former US presidential candidates.

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Notes

  1. 1.

    The partial volume effect is responsible for the reduced accuracy with which the boundaries of small formations can be determined in clinical images.

  2. 2.

    This analysis could not be performed for the incumbent president of the USA, Donald J. Trump, since his tax return for the year immediately preceding his inauguration was not made available.

References

  1. 1.

    Arshadi, L., Jahangir, A.H.: Benford’s law behavior of Internet traffic. J. Netw. Comput. Appl. 40, 194–205 (2014)

    Article  Google Scholar 

  2. 2.

    Benford, F.: The law of anomalous numbers. Proc. Am. Philos. Soc. 78, 551–572 (1938)

    MATH  Google Scholar 

  3. 3.

    Corazza, M., Ellero, A., Zorzi, A.: Checking financial markets via Benford’s law: The S&P 5000 case. In: Corazza, M., Pizzi, C. (eds.) Mathematical and Statistical Methods for Actuarial Sciences and Finance, pp. 93–102. Springer, Berlin (2010)

    Google Scholar 

  4. 4.

    Diaconis, P., Freedman, D.: On rounding percentages. J. Am. Stat. Assoc. 74, 359–364 (1979)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Diekmann, A.: Not the first digit! Using Benford’s law to detect fraudulent scientific data. J. Appl. Stat. 37, 321–329 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Gauvrit, N., Delahaye, J.-P.: Pourquoi la loi de Benford n’est pas mystérieuse. Mathématiques et Sciences Humaines 182, 7–15 (2008)

    Article  MATH  Google Scholar 

  7. 7.

    Hamming, R.: On the distribution of numbers. Bell Syst. Tech. J. 49, 1609–1625 (1970)

    Article  MATH  Google Scholar 

  8. 8.

    Hill, T.P.: A statistical derivation of the significant-digit law. Stat. Sci. 10, 354–363 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Hill, T.P.: The significant-digit phenomenon. Am. Math. Mon. 102, 322–327 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Miller S.J. (editor): Benford’s Law: Theory and Applications. Princeton University Press (2015)

  11. 11.

    Newcomb, S.: Note on the frequency of use of the different digits in natural numbers. Am. J. Math. 4, 39–40 (1881)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Nigrini, M.: A taxpayer compliance application of Benford’s law. J. Am. Tax. Assoc. 18, 72–91 (1996)

    Google Scholar 

  13. 13.

    Nigrini, M.: I’ve got your number: How a mathematical phenomenon can help CPAs uncover fraud and other irregularities. J. Account. 79–83 (1999)

  14. 14.

    Orita, M., Moritomo, A., Niimi, T., Ohno, K.: Use of Benford’s law in drug discovery data. Drug Discov. Today 15, 328–331 (2010)

    Article  Google Scholar 

  15. 15.

    Pérez-González, F., Abdallah, C.T., Heileman, G.L.: Benford’s law in image processing. IEEE Int Conf Image Process 405–408 (2007)

  16. 16.

    Phillips, T.: Simon Newcomb and “Natural Numbers” (Benford’s law). American Mathematical Society Feature Column—Monthly Essays on Mathematical Topics (2009). http://www.ams.org

  17. 17.

    Roukema, B.F.: A first-digit anomaly in the 2009 Iranian presidential election. J. Appl. Stat. 41, 164–199 (2014)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Wells, K., Chiverton, J., Partridge, M., Barry, M., Kadhem, H., Ott, B.: Quantifying the partial volume effect in PET using Benford’s law. IEEE Trans. Nucl. Sci. 54, 1616–1625 (2007)

    Article  Google Scholar 

  19. 19.

    Zorzi, A.: An elementary proof for the Equidistribution Theorem. Math Intell 37, 1–2 (2015)

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Marco Corazza.

Additional information

The authors thank the Centro di Economia Quantitativa of the Ca’ Foscari University of Venice (Italy) for the support received.

Appendix: Benford’s law

Appendix: Benford’s law

Benford’s law, in its more complete form, claims that the probability that the i-th digit \(D_i\) equals \(d_i\) for \(i = 1, \ldots , n\), that is, for the first n base-10 digits, is

$$\begin{aligned} \log _{10}\left( 1 + \left( \sum \limits _{i=1}^{n} 10^{n-i}d_i \right) ^{-1}\right) . \end{aligned}$$

Moreover, Benford’s law has several interesting characterisations.

First, a sequence \((x_n)_n\) satisfies Benford’s law if and only if the sequence \((\{log_{10}\left| x_n\right| \})_n\) is equidistributed in the interval [0, 1), where \(\{x \}\) is the fractional part of x.

Then, when the numbers we are considering are given by physical measurements, Benford’s law does not depend on the units chosen (scale invariance). Moreover, it does not depend on the base chosen to represent the numbers either (base invariance).

For further information we refer the interested reader to Theodore P. Hill’s article [9].

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Corazza, M., Ellero, A. & Zorzi, A. The importance of being “one” (or Benford’s law). Lett Mat Int 6, 33–39 (2018). https://doi.org/10.1007/s40329-018-0218-4

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Keywords

  • Benford’s law
  • First significant digit
  • Power sequences
  • Tax evasion
  • Vote-rigging