Involve: A Journal of Mathematics

  • Involve
  • Volume 8, Number 5 (2015), 859-874.

The Weibull distribution and Benford's law

Victoria Cuff, Allison Lewis, and Steven J. Miller

Full-text: Open access


Benford’s law states that many data sets have a bias towards lower leading digits (about 30% are 1s). It has numerous applications, from designing efficient computers to detecting tax, voter and image fraud. It’s important to know which common probability distributions are almost Benford. We show that the Weibull distribution, for many values of its parameters, is close to Benford’s law, quantifying the deviations. As the Weibull distribution arises in many problems, especially survival analysis, our results provide additional arguments for the prevalence of Benford behavior. The proof is by Poisson summation, a powerful technique to attack such problems.

Article information

Involve, Volume 8, Number 5 (2015), 859-874.

Received: 31 July 2014
Revised: 19 October 2014
Accepted: 1 December 2014
First available in Project Euclid: 22 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 11K06: General theory of distribution modulo 1 [See also 11J71]
Secondary: 60E10: Characteristic functions; other transforms 42A16: Fourier coefficients, Fourier series of functions with special properties, special Fourier series {For automorphic theory, see mainly 11F30} 62E15: Exact distribution theory 62P99: None of the above, but in this section

Benford's law Weibull distribution digit bias Poisson summation


Cuff, Victoria; Lewis, Allison; Miller, Steven J. The Weibull distribution and Benford's law. Involve 8 (2015), no. 5, 859--874. doi:10.2140/involve.2015.8.859.

Export citation


  • C. T. Abdallah, G. L. Heileman, S. J. Miller, F. Pérez-González, and T. Quach, “Application of Benford's law to images”, in Theory and applications of Benford's law, edited by S. J. Miller, Princeton University Press, 2015.
  • M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series 55, U.S. Government Printing Office, Washington, DC, 1964. Reprinted by Dover, New York, 1974.
  • L. Arshadi and A. H. Jahangir, “Benford's law behavior of internet traffic”, J. Netw. Comput. Appl. 40 (2014), 194–205.
  • F. Benford, “The law of anomalous numbers”, Proc. Amer. Phil. Soc. 78:4 (1938), 551–572.
  • A. Berger and T. P. Hill, “Fundamental flaws in Feller's classical derivation of Benford's law”, preprint, 2010.
  • A. Berger and T. P. Hill, “A basic theory of Benford's law”, Probab. Surv. 8 (2011), 1–126.
  • A. Berger and T. P. Hill, “Benford's law strikes back: no simple explanation in sight for mathematical gem”, Math. Intelligencer 33:1 (2011), 85–91.
  • A. Berger and T. P. Hill, “Benford online bibliography”, 2012,
  • K. J. Carroll, “On the use and utility of the Weibull model in the analysis of survival data”, Control. Clin. Trials 24:6 (2003), 682–701.
  • W. K. T. Cho and B. J. Gaines, “Breaking the (Benford) law: statistical fraud detection in campaign finance”, Amer. Stat. 61:3 (2007), 218–223.
  • O. Corzo and N. Bracho, “Application of Weibull distribution model to describe the vacuum pulse osmotic dehydration of sardine sheets”, LWT Food Sci. Technol. 41:6 (2008), 1108–1115.
  • P. Diaconis, “The distribution of leading digits and uniform distribution ${\rm mod}$ $1$”, Ann. Probability 5:1 (1977), 72–81.
  • L. Dümbgen and C. Leuenberger, “Explicit bounds for the approximation error in Benford's law”, Electron. Commun. Probab. 13 (2008), 99–112.
  • W. Feller, An introduction to probability theory and its applications, vol. 2, 2nd ed., Wiley, New York, 1966.
  • R. M. Fewster, “A simple explanation of Benford's law”, Amer. Stat. 63:1 (2009), 26–32.
  • S. Fry, “How political rhetoric contributes to the stability of coercive rule: a Weibull model of post-abuse government survival”, 2004. Paper presented at the annual meeting of the International Studies Association (Montreal, 2004).
  • T. P. Hill, “A statistical derivation of the significant-digit law”, Statist. Sci. 10:4 (1995), 354–363.
  • T. P. Hill, “The first digit phenomenon”, Amer. Sci. 86:4 (1998), 358–363.
  • T. P. Hill, “Benford's law blunders”, Amer. Stat. 65:2 (2011), 141.
  • D. Jang, J. U. Kang, A. Kruckman, J. Kudo, and S. J. Miller, “Chains of distributions, hierarchical Bayesian models and Benford's law”, J. Alg. Number Theory Adv. Appl. 1:1 (2009), 37–60.
  • G. Judge and L. Schechter, “Detecting problems in survey data using Benford's law”, J. Hum. Resour. 44:1 (2009), 1–24.
  • D. E. Knuth, The art of computer programming, 2: Seminumerical algorithms, 3rd ed., Addison-Wesley, Reading, MA, 1997.
  • A. V. Kontorovich and S. J. Miller, “Benford's law, values of $L$-functions and the $3x+1$ problem”, Acta Arith. 120:3 (2005), 269–297.
  • L. M. Leemis, B. W. Schmeiser, and D. L. Evans, “Survival distributions satisfying Benford's law”, Amer. Stat. 54:4 (2000), 236–241.
  • B. McShane, M. Adrian, E. T. Bradlow, and P. S. Fader, “Count models based on Weibull interarrival times”, J. Bus. Econom. Statist. 26:3 (2008), 369–378.
  • W. R. Mebane, Jr., “Election forensics: the second-digit Benford's law test and recent American presidential elections”, 2006, Paper presented at the Election Fraud Conference (Salt Lake City, UT, 2006).
  • P. G. Mikolaj, “Environmental applications of the Weibull distribution function: oil pollution”, Science 176:4038 (1972), 1019–1021.
  • S. J. Miller, “A derivation of the Pythagorean won-loss formula in baseball”, Chance 20:1 (2007), 40–48.
  • S. J. Miller (editor), Benford's law: theory and applications, Princeton University Press, 2015.
  • S. J. Miller and M. J. Nigrini, “The modulo $1$ central limit theorem and Benford's law for products”, Int. J. Algebra 2:3 (2008), 119–130.
  • S. J. Miller and M. J. Nigrini, “Order statistics and Benford's law”, Int. J. Math. Math. Sci. 2008 (2008), Article ID 382948.
  • S. J. Miller and M. J. Nigrini, “Data diagnostics using second order tests of Benford's Law”, Audit. J. Pract. Theory 28:2 (2009), 305–324.
  • S. J. Miller and R. Takloo-Bighash, An invitation to modern number theory, Princeton University Press, 2006.
  • S. Newcomb, “Note on the frequency of use of the different digits in natural numbers”, Amer. J. Math. 4:1-4 (1881), 39–40.
  • M. J. Nigrini, “Digital analysis and the reduction of auditor litigation risk”, pp. 68–81 in Auditing symposium XIII: proceedings of the 1996 Deloitte & Touche/University of Kansas Symposium on Auditing Problems (Lawrence, KS, 1996), edited by M. L. Ettredge, Division of Accounting and Information Systems, School of Business, University of Kansas, Lawrence, KS, 1996.
  • M. J. Nigrini, “The use of Benford's law as an aid in analytical procedures”, Audit. J. Pract. Theory 16:2 (1997), 52–67.
  • R. A. Raimi, “The first digit problem”, Amer. Math. Monthly 83:7 (1976), 521–538.
  • E. M. Stein and R. Shakarchi, Fourier analysis: an introduction, Princeton Lectures in Analysis 1, Princeton University Press, 2003.
  • Y. Terawaki, T. Katsumi, and V. Ducrocq, “Development of a survival model with piecewise Weibull baselines for the analysis of length of productive life of Holstein cows in Japan”, J. Dairy Sci. 89:10 (2006), 4058–4065.
  • W. Weibull, “A statistical distribution function of wide applicability”, J. Appl. Mech. 18 (1951), 293–297.
  • E. T. Whittaker and G. N. Watson, A course of modern analysis: an introduction to the general theory of infinite processes and of analytic functions, with an account of the principal transcendental functions, Reprint of the 4th ed., Cambridge University Press, 1996.
  • C. T. Yiannoutsos, “Modeling AIDS survival after initiation of antiretroviral treatment by Weibull models with changepoints”, J. Int. AIDS Soc. 12:9 (2009).
  • Y. Zhao, A. H. Lee, K. K. W. Yau, and G. J. McLachlan, “Assessing the adequacy of Weibull survival models: a simulated envelope approach”, J. Appl. Stat. 38:10 (2011), 2089–2097.