Benford’s Law: Textbook Exercises and Multiple-Choice Testbanks

Slepkov, Aaron D.; Ironside, Kevin B.; DiBattista, David

doi:10.1371/journal.pone.0117972

Cited by 20 publications

(13 citation statements)

References 25 publications

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“…In this study, we further apply additional tests, i.e., Mean Absolute Deviation (MAD) and Sum of Squared Difference (SSD), which are less dependent on the sample size [20][21][22]. The Equation (4) shows the MAD calculation as the average absolute deviation of observed and expected frequencies:…”

Section: Benford's Law and Goodness-of-fit Testsmentioning

confidence: 99%

Are COVID-19 Data Reliable? A Quantitative Analysis of Pandemic Data from 182 Countries

Farhadi

Lahooti

2021

COVID

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When it comes to COVID-19, access to reliable data is vital. It is crucial for the scientific community to use data reported by independent territories worldwide. This study evaluates the reliability of the pandemic data disclosed by 182 countries worldwide. We collected and assessed conformity of COVID-19 daily infections, deaths, tests, and vaccinations with Benford’s law since the beginning of the coronavirus pandemic. It is commonly accepted that the frequency of leading digits of the pandemic data shall conform to Benford’s law. Our analysis of Benfordness elicits that most countries partially distributed reliable data over the past eighteen months. Notably, the UK, Australia, Spain, Israel, and Germany, followed by 22 different nations, provided the most reliable COVID-19 data within the same period. In contrast, twenty-six nations, including Tajikistan, Belarus, Bangladesh, and Myanmar, published less reliable data on the coronavirus spread. In this context, over 31% of countries worldwide seem to have improved reliability. Our measurement of Benfordness moderately correlates with Johns Hopkin’s Global Health Security Index, suggesting that the quality of data may depend on national healthcare policies and systems. We conclude that economically or politically distressed societies have declined in conformity to the law over time. Our results are particularly relevant for policymakers worldwide.

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Section: Benford's Law and Goodness-of-fit Testsmentioning

confidence: 99%

Are COVID-19 Data Reliable? A Quantitative Analysis of Pandemic Data from 182 Countries

Farhadi

Lahooti

2021

COVID

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“…The results of these 14 Monte Carlo schemes are given in the table of Figure 26 and its associated bar chart of Figure 27. Increasingly, scholars apply both SSD as well as MAD in their Benford's Law research, such as in [10][11][12].…”

Section: Testing the Lognormal Via The Chi-square Statisticmentioning

confidence: 99%

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

Kossovsky¹

2021

Stats

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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.

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“…Specifically, letting be country i ’s confirmed case number at time t in base b , for t = 1, 2, …, T , we compare the leading digit frequencies observed up to time T to the expected leading digit frequencies under the assumption that Benford’s law holds. This is one of the tests used by Deleanu [ 8 ] who studies the applicability of using Benford’s law to detect false criminal enforcement statistics, and is noted in [ 22 ] as being the most common goodness of fit test used to assess the validity of Benford’s law.…”

Section: Benford’s Lawmentioning

confidence: 99%

On the authenticity of COVID-19 case figures

Kennedy

Yam

2020

PLoS ONE

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In this article, we study the applicability of Benford’s law and Zipf’s law to national COVID-19 case figures with the aim of establishing guidelines upon which methods of fraud detection in epidemiology, based on formal statistical analysis, can be developed. Moreover, these approaches may also be used in evaluating the performance of public health surveillance systems. We provide theoretical arguments for why the empirical laws should hold in the early stages of an epidemic, along with preliminary empirical evidence in support of these claims. Based on data published by the World Health Organization and various national governments, we find empirical evidence that suggests that both Benford’s law and Zipf’s law largely hold across countries, and deviations can be readily explained. To the best of our knowledge, this paper is among the first to present a practical application of Zipf’s law to fraud detection.

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Benford’s Law: Textbook Exercises and Multiple-Choice Testbanks

Cited by 20 publications

References 25 publications

Are COVID-19 Data Reliable? A Quantitative Analysis of Pandemic Data from 182 Countries

Are COVID-19 Data Reliable? A Quantitative Analysis of Pandemic Data from 182 Countries

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

On the authenticity of COVID-19 case figures

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