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Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf

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  • Pietronero, L.
  • Tosatti, E.
  • Tosatti, V.
  • Vespignani, A.
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    Abstract

    The distribution of first digits in numbers series obtained from very different origins shows a marked asymmetry in favor of small digits that goes under the name of Benford's law. We analyze in detail this property for different data sets and give a general explanation for the origin of the Benford's law in terms of multiplicative processes. We show that this law can be also generalized to series of numbers generated from more complex systems like the catalogs of seismic activity. Finally, we derive a relation between the generalized Benford's law and the popular Zipf's law which characterize the rank order statistics and has been extensively applied to many problems ranging from city population to linguistics.

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    File URL: http://www.sciencedirect.com/science/article/pii/S0378437100006336
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    Bibliographic Info

    Article provided by Elsevier in its journal Physica A: Statistical Mechanics and its Applications.

    Volume (Year): 293 (2001)
    Issue (Month): 1 ()
    Pages: 297-304

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    Handle: RePEc:eee:phsmap:v:293:y:2001:i:1:p:297-304

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    Web page: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/

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    Cited by:
    1. Mir, T.A., 2014. "The Benford law behavior of the religious activity data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 408(C), pages 1-9.
    2. Tariq Ahmad Mir, 2012. "The leading digit distribution of the worldwide Illicit Financial Flows," Papers 1201.3432, arXiv.org, revised Nov 2012.
    3. Ferrer i Cancho, Ramon, 2005. "Decoding least effort and scaling in signal frequency distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 345(1), pages 275-284.
    4. Dorogovtsev, S.N. & Mendes, J.F.F. & Oliveira, J.G., 2006. "Frequency of occurrence of numbers in the World Wide Web," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 360(2), pages 548-556.
    5. Lee, Joanne & Cho, Wendy K. Tam & Judge, George G., 2010. "Stigler's approach to recovering the distribution of first significant digits in natural data sets," Statistics & Probability Letters, Elsevier, vol. 80(2), pages 82-88, January.
    6. Paulette Clippe & Marcel Ausloos, 2012. "Benford's law and Theil transform of financial data," Papers 1208.5896, arXiv.org.
    7. Lee, Joanne & Cho, Wendy K. Tam & Judge, George G, 2009. "Stigler's approach to recovering the distribution of first significant digits in natural data sets," Department of Agricultural & Resource Economics, UC Berkeley, Working Paper Series qt9745m98d, Department of Agricultural & Resource Economics, UC Berkeley.
    8. Lee, Joanne & Judge, George G, 2008. "Identifying falsified clinical data," Department of Agricultural & Resource Economics, UC Berkeley, Working Paper Series qt8x00h1c1, Department of Agricultural & Resource Economics, UC Berkeley.
    9. David E. Giles, 2005. "Benford’s Law and Naturally Occurring Prices in Certain ebaY Auctions," Econometrics Working Papers 0505, Department of Economics, University of Victoria.

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