LIMITS OF BENFORD'S LAW IN EXPERIMENTAL FIELD
Stéphane Blondeau Da Silva
13 François Chénieux Street
Limoges 87000, FRANCE

Abstract

Two of the main flaws of Benford's law will be discussed in this article: (i) the first one, which leads the observer to consider an experimental dataset as the result of a single random variable rather than several, makes this law so mysterious; (ii) the second one is that Benford's probabilities have long been considered as perfect values: this is obviously not the case.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 33
Issue: 4
Year: 2020

DOI: 10.12732/ijam.v33i4.12

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References

  1. [1] T. Alexopoulos, S. Leontsinis, Benford’s law in astronomy, Journal of Astrophysics and Astronomy, 35, No 4 (2014), 639-648.
  2. [2] A. D. Alves, H. H. Yanasse, N. Y. Soma, Benford’s law and articles of scientific journals: comparison of JCR and Scopus data, Scientometrics, 98 (2014), 173-184.
  3. [3] J. Aron, Crime-fighting maths law confirms planetary riches, New Scientist, 220 (2013), 12-13.
  4. [4] L. Arshadi, A. H. Jahangir, Benford’s law behavior of internet traffic, Journal of Network and Computer Applications, 40 (2014), 194-205.
  5. [5] E. P. Balanzario, J. Snchez-Ortiz, Sufficient conditions for Benford’s law, Statistics ans Probability Letters, 80, No 23-24 (2010), 1713-1719.
  6. [6] G. Beeli, M. Esslen, L. Jncke, Frequency correlates in grapheme-color synæsthesia, Psychological Science, 18 (2007), 788-792.
  7. [7] T. W. Beer, Terminal digit preference: beware of Benford’s law, Journal of Clinical Pathology, 62, No 2 (2009), 192.
  8. [8] F. Benford, The law of anomalous numbers, Proceedings of the American Philosophical Society, 78 (1938), 127-131.
  9. [9] A. Berger, T. P. Hill, Benford’s law strikes back: No simple explanation in sight for mathematical gem, The Mathematical Intelligencer, 33, No 1 (2011), 85-91.
  10. [10] A. Berger, T. P. Hill, An Introduction to Benford’s Law, Princeton University Press, Princeton (2015).
  11. [11] S. Blondeau Da Silva, Benford or not Benford: a systematic but not always well-founded use of an elegant law in experimental fields, Communications in Mathematics and Statistics, 8 (2020), 167-201.
  12. [12] A. Bonache, J. Maurice, K. Moris, Dtection de fraudes et loi de Benford : Quelques risques associes, Revue Franaise de Comptabilit, 431 (2010), 24-27.
  13. [13] J. Burke, E. Kincanon, Benford’s law and physical constants: the distribution of initial digits, American Journal of Physics, 59 (1991), 952.
  14. [14] R. S. Cella, E. Zanolla, Benford’s law and transparency: an analysis of municipal expenditure, Brazilian Business Review, 15, No 4 (2018), 331347.
  15. [15] M. C. Chou, Q. Kong, C. P. Teo, Z. Wang, H. Zheng, Benford’s law and number selection in fixed-odds numbers game, Journal of Gambling Studies, 25 (2009), 503-521.
  16. [16] P. Clippe, M. Ausloos, Benford’s law and theil transform of financial data, Physica A: Statistical Mechanics and its Applications, 391, No 4 (2012), 6556-6567.
  17. [17] E. Costasa, V. Lopez-Rodasa, F. Torob, A. Flores-Moya, The number of cells in colonies of the cyanobacterium microcystis aeruginosa satisfies Benford’s law, Aquatic Botany, 89, No 3 (2008), 341-343.
  18. [18] S. Cournane, N. Sheehy, J. Cooke, The novel application of Benford’s second order analysis for monitoring radiation output in interventional radiology, Physica Medica, 30 (2014), 413-418.
  19. [19] J. Deckert, M. Myagkov, P. Ordeshook, Benford’s law and the detection of election fraud, Political Analysis, 19 (2011), 245-268.
  20. [20] S. Dehaene, J. Mehler, Cross-linguistic regularities in the frequency of number words, Cognition, 43 (1992), 1-29.
  21. [21] A. Diekmann, B. Jann, Benford’s law and fraud detection: Facts and legends, German Economic Review, 11, No 3 (2010), 397-401.
  22. [22] C. Durtschi, W. Hillison, C. Pacini, The effective use of Benford’s law to assist in detecting fraud in accounting data, Journal of Forensic Accounting, V (2004), 17-34.
  23. [23] R. M. Fewster, A simple explanation of Benford’s law, The American Statistician, 63, No 1 (2009), 26-32.
  24. [24] B. J. Flehinger, On the probability that a random integer has initial digit a, American Mathematical Monthly, 73, No 10 (1966), 1056-1061.
  25. [25] A. K. Formann, The Newcomb-Benford law in its relation to some common distributions, Plos One, 5 (2010).
  26. [26] J. L. Friar, T. Goldman, J. Prez-Mercader, Genome sizes and the Benford distribution, Plos One, 7, No 5 (2012).
  27. [27] N. Gauvrit, J.-P. Delahaye, Scatter and Regularity Imply Benford’s Law... and More, World Scientific (2011), 53-69.
  28. [28] J. Golbeck, Benford’s law applies to online social networks, Plos One, 10, No 8 (2015).
  29. [29] W. Goodman, The promises and pitfalls of Benford’s law, Significance, 13, No 3 (2016), 38-41.
  30. [30] A. Herzel, Sulla distribuzione delle cifre iniziali dei numeri statistic, In: Atti XV e XVI Riunione Sci., Roma (1956), 205-228.
  31. [31] M. J. Hickman, S. K. Rice, Digital analysis of crime statistics: does crime conform to Benford’s law?, Journal of Quantitative Criminology, 26 (2010), 333-349.
  32. [32] T. P. Hill, Random-number guessing and the first digit phenomenon, Psychological Reports, 62, No 3 (1988), 967-971.
  33. [33] T. P. Hill, A statistical derivation of the Significant-Digit law, Statistical Science, 10, No 4 (1995), 354-363.
  34. [34] T. P. Hill, The first digit phenomenon: A century-old observation about an unexpected pattern in many numerical tables applies to the stock market, census statistics and accounting data, American Scientist, 86, No 4 (1998), 358-363.
  35. [35] E. Janvresse, T. De La Rue, From uniform distributions to Benford’s law, Journal of Applied Probability, 41, No 4 (2004), 1203-1210.
  36. [36] D. Knuth, The Art of Computer Programming 2, Addison-Wesley, NewYork (1969).
  37. [37] M. Kreuzer, D. Jordan, B. Antkowiak, B. Drexler, E. F. Kochs, G. Schneider, Brain electrical activity obeys Benford’s law, Anesthesia & Analgesia, 118 (2014), 183-191.
  38. [38] L. Lacasa, J. Fernandez-Gracia, Election forensics: Quantitative methods for electoral fraud detection, Forensic Science International, 294 (2019), e19-e22.
  39. [39] L. Leemis, B. Schmeiser, D. Evans, Survival distributions satisfying Benford’s law, The American Statistician, 54, No 4 (2000), 236-241.
  40. [40] T. Lolbert, On the non-existence of a general Benford’s law, Mathematical Social Sciences, 55 (2008), 103-106.
  41. [41] S. J. Miller, Benford’s Law: Theory and Applications, Princeton University Press, Princeton (2015).
  42. [42] T. A. Mir, The Benford law behavior of the religious activity data, Physica A: Statistical Mechanics and its Applications, 408 (2014), 1-9.
  43. [43] R. Newcomb, Note on the frequency of use of the different digits in natural numbers, American Journal of Mathematics, 4 (1881), 39-40.
  44. [44] R. E. J. Newman, Power laws, Pareto distributions and Zipf’s law, Contemporary Physics, 46, No 5 (2005), 323-351.
  45. [45] M. Nigrini, I’ve got your number: How a mathematical phenomenon can help CPAs uncover fraud and other irregularities, Journal of Accountancy, (1999).
  46. [46] M. Nigrini, Benford’s Law: Applications for Forensic Accounting, Auditing, and Fraud Detection, Vol. 586, John Wiley & Sons, (2012).
  47. [47] M. Nigrini, S. Miller, Benford’s law applied to hydrology data-results and relevance to other geophysical data, Mathematical Geology, 39, No 5 (2007), 469-490.
  48. [48] M. Nigrini, W. Wood, Assessing the integrity of tabulated demographic data, Preprint (1995).
  49. [49] T. Piketty, Le Capital au XXIe sicle, Vol. 585, Seuil (2013).
  50. [50] R. A. Raimi, The first digit problem, American Mathematical Monthly, 83, No 7 (1976), 521-538.
  51. [51] B. Rauch, M. Gttsche, G. Brlher, S. Engel, Fact and fiction in EUgovernmental economic data, German Economic Review, 12, No 3 (2011), 243-255.
  52. [52] M. Sambridge, H. Tkalcic, P. Arroucau, Benford’s law of first digits: From mathematical curiosity to change detector, Asia Pacific Mathematics Newsletter, 1 (2011), 1-6.
  53. [53] A. Saville, Using Benford’s law to detect data error and fraud: An examination of companies listed on the Johannesburg Stock Exchange, South African Journal of Economic and Management Sciences, 9, No 3 (2006), 341-354.
  54. [54] P. D. Scott, M. Fasli, Benford’s Law: An empirical investigation and a novel explanation, In: CSM Technical Report 349, University of Essex (2001).
  55. [55] T. Sehity, E. Hoelz, E. Kirchler, Price developments after a nominal shock: Benford’s law and psychological pricing after the euro introduction, International Journal of Research in Marketing, 22, No 4 (2005), 471-480.
  56. [56] S. W. Smith, The Scientist and Engineer’s Guide to Digital Signal Processing, Ch. 34: Explaining Benford’s law (2012).
  57. [57] C. Tolle, J. Budzien, R. Laviolette, Do dynamical systems follow Benford’s law?, Chaos, 10, No 2 (2000).
  58. [58] K. Tdter, Benford’s law as an indicator of fraud in economics, German Economic Review, 10 (2009), 339-351.
  59. [59] H. Varian, Benford’s law, The American Statistician, Letters to the Editor, 26, No 3 (1972), 62-65.
  60. [60] G. Whyman, E. Shulzinger, E. Bormashenko, Intuitive considerations clarifying the origin and applicability of the Benford law, Results in Physics, 6 (2016), 3-6.