Benford’s Law and Sum Invariance Testing

  • Zoran JasakEmail author
Conference paper
Part of the Lecture Notes in Networks and Systems book series (LNNS, volume 59)


Benford’s law is logarithmic law for distribution of leading digits formulated by P[D = d] = log(1 + 1/d) where d is leading digit or group of digits. It’s named by Frank Albert Benford (1938) who formulated mathematical model of this probability. Before him, the same observation was made by Simon Newcomb. This law has changed usual preasumption of equal probability of each digit on each position in number. One of main characteristic properties of this law is sum invariance. Sum invariance means that sums of significand are the same for any leading digit or group of digits. Term ‘significand’ is used instead of term ‘mantissa’ to avoid terminological confusion with logarithmic mantissa.



I wish to thank to Mr. Wilhelm Schappacher for great support in my work.


  1. 1.
    Newcomb, S.: Note on the frequency of use of different digits in natural numbers. Am. J. Math. 4, 39–40 (1881)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Benford, F.A.: The law of anomalous numbers. Proc. Am. Philos. Soc. 78, 551–572 (1938)zbMATHGoogle Scholar
  3. 3.
    Strauch, O.: Unsolved problems. Tatra Mt. Math. Publ. 56(3), 175–178 (2013)zbMATHGoogle Scholar
  4. 4.
    Berger, A., Hill, T.P.: Theory of Benford’s Law. Probab. Surv. 8, 1–126 (2011). ISSN 1549-5787MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hill, T.P.: Base invariance implies Benford’s Law. Proc. Am. Math. Soc. 123(3), 887–895 (1995)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Nigrini, M.: Forensic Analytics – Methods and Techniques for Forensic Accounting Investigations, pp. 144–146. Wiley, Hoboken (2011)CrossRefGoogle Scholar
  7. 7.
    Allart, P.C.: A Sum-invariant Charcterization of Benford’s Law. AMS (1990)Google Scholar
  8. 8.
    Dumas, C., Devine, J.S.: Detecting evidence of non-compliance in self-reported pollution emissions data: an application of Benford’s law. Selected Paper American Agricultural Economics Association Annual Meeting Tampa, Fl, 30 July–2 August 2000Google Scholar
  9. 9.
    Salicru, M., Morales, D., Menendez, M.L., Pardo, L.: On the Application of Divergence Type Measures in Testing Statistical Hypotheses. J. Multivar. Anal. 51, 372–391 (1994)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jasak, Z.: Sum invariance testing and some new properties of Benford’s law, Doctorial dissertation. University in Tuzla, Bosnia and Herzegovina (2017)Google Scholar
  11. 11.
    Jasak, Z.: Benford’s law and invariances. J. Math. Syst. Sci. 1(1), 1–6 (2011). (Serial No.1). ISSN 2159-5291Google Scholar
  12. 12.
    Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22(1), 79–86 (1951)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.NLB Banka d.d.SarajevoBosnia and Herzegovina

Personalised recommendations