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On the Mistaken Use of the Chi-Square Test in Benford’s Law

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  • Alex Ely Kossovsky

    (Independent Researcher, New York, NY 10002, USA)

Abstract

Benford’s Law predicts that the first significant digit on the leftmost side of numbers in real-life data is distributed between all possible 1 to 9 digits approximately as in LOG(1 + 1/digit), so that low digits occur much more frequently than high digits in the first place. Typically researchers, data analysts, and statisticians, rush to apply the chi-square test in order to verify compliance or deviation from this statistical law. In almost all cases of real-life data this approach is mistaken and without mathematical-statistics basis, yet it had become a dogma or rather an impulsive ritual in the field of Benford’s Law to apply the chi-square test for whatever data set the researcher is considering, regardless of its true applicability. The mistaken use of the chi-square test has led to much confusion and many errors, and has done a lot in general to undermine trust and confidence in the whole discipline of Benford’s Law. This article is an attempt to correct course and bring rationality and order to a field which had demonstrated harmony and consistency in all of its results, manifestations, and explanations. The first research question of this article demonstrates that real-life data sets typically do not arise from random and independent selections of data points from some larger universe of parental data as the chi-square approach supposes, and this conclusion is arrived at by examining how several real-life data sets are formed and obtained. The second research question demonstrates that the chi-square approach is actually all about the reasonableness of the random selection process and the Benford status of that parental universe of data and not solely about the Benford status of the data set under consideration, since the focus of the chi-square test is exclusively on whether the entire process of data selection was probable or too rare. In addition, a comparison of the chi-square statistic with the Sum of Squared Deviations (SSD) measure of distance from Benford is explored in this article, pitting one measure against the other, and concluding with a strong preference for the SSD measure.

Suggested Citation

  • Alex Ely Kossovsky, 2021. "On the Mistaken Use of the Chi-Square Test in Benford’s Law," Stats, MDPI, vol. 4(2), pages 1-35, May.
    • Handle: RePEc:gam:jstats:v:4:y:2021:i:2:p:27-453:d:563997
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    References listed on IDEAS

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    1. Cerqueti, Roy & Maggi, Mario, 2021. "Data validity and statistical conformity with Benford’s Law," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
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    Cited by:

    1. Roy Cerqueti & Claudio Lupi, 2023. "Severe testing of Benford’s law," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(2), pages 677-694, June.
    2. Arezzo, Maria Felice & Cerqueti, Roy, 2023. "A Benford’s Law view of inspections’ reasonability," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 632(P1).
    3. Sylwestrzak Marek, 2023. "Applying Benford’s law to detect earnings management," Journal of Economics and Management, Sciendo, vol. 45(1), pages 216-236, January.
    4. Eutsler, Jared & Kathleen Harris, M. & Tyler Williams, L. & Cornejo, Omar E., 2023. "Accounting for partisanship and politicization: Employing Benford's Law to examine misreporting of COVID-19 infection cases and deaths in the United States," Accounting, Organizations and Society, Elsevier, vol. 108(C).

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