scholarly journals Comment on “Benford's Law and the Detection of Election Fraud”

2011 ◽  
Vol 19 (3) ◽  
pp. 269-272 ◽  
Author(s):  
Walter R. Mebane
Keyword(s):  
Careful Analysis ◽  
Benford’S Law ◽  
Simulation Exercise ◽  
Strategic Actions ◽  
Significant Digit ◽  
Benford's Law ◽  
The Mean ◽  
Election Fraud ◽  
Open Question

“Benford's Law and the Detection of Election Fraud” raises doubts about whether a test based on the mean of the second significant digit of vote counts equals 4.187 is useful as a test for the occurrence of election fraud. The paper mistakenly associates such a test with Benford's Law, considers a simulation exercise that has no apparent relevance for any actual election, applies the test to inappropriate levels of aggregation, and ignores existing analysis of recent elections in Russia. If tests based on the second significant digit of precinct-level vote counts are diagnostic of election fraud, the tests need to use expectations that take into account the features of ordinary elections, such as strategic actions. Whether the tests are useful for detecting fraud remains an open question, but approaching this question requires an approach more nuanced and tied to careful analysis of real election data than one sees in the discussed paper.

scholarly journals Benford’s Law Application: Case of Elections in Sri Lanka

2020 ◽  
Author(s):  
Richmond Sam Quarm ◽  
Richmond Sam-Quarm
Keyword(s):  
Sri Lanka ◽  
Political Parties ◽  
Presidential Election ◽  
Benford’S Law ◽  
Third Place ◽  
Benford's Law ◽  
Data Points ◽  
The Mean ◽  
Election Fraud ◽  
Election Results

Sri Lanka, like many developing countries has been involved in a circle of allegations of election fraud. Usually these claims are pronounced more by losing parties. This study uses Benford’s law, a law of probability distribution of digits, to investigate whether the election fraud claims might have merit. A sample in this study is made of 808 election data. This data comes from the 2010 Presidential election for representatives from three major political parties and from 2010 General Election data. All of the data points were obtained through reliable government sources, two of which are, the Department of Elections website of Sri Lanka, and the National News Paper statistics (2010). The study contrasts the distribution of the first digit of election results against the Benford’s Law benchmark. After obtaining the results, we organize the data and find median, mean, mode and standard deviation. The preliminary results showe that the data does not align with Benford’s law predictions. In other words, it shows that the data does not follow the law where the mean is larger than the median and there is a positive skewness then it likely follows a Benford’s distribution. The distribution of the first digit of actual data for three parties disagrees with Benford's law. This misalignment is more pronounced for the winning party than for the second and third place parties, respectively. We, therefore, look forward to run the data through several critical analyses and observing if there shall be any fraud or manipulation in numbers.

scholarly journals Benford’s Law Application: Case of Elections in Sri Lanka

2020 ◽  
Vol 3 (4) ◽  
Author(s):  
Agim Kukeli ◽  
◽  
Hettiyadura Shehan Karunaratne ◽  
Keyword(s):  
Sri Lanka ◽  
Political Parties ◽  
Presidential Election ◽  
Benford’S Law ◽  
Third Place ◽  
Benford's Law ◽  
Data Points ◽  
The Mean ◽  
Election Fraud ◽  
Election Results

Sri Lanka, like many developing countries has been involved in a circle of allegations of election fraud. Usually these claims are pronounced more by losing parties. This study uses Benford’s law, a law of probability distribution of digits, to investigate whether the election fraud claims might have merit. A sample in this study is made of 808 election data. This data comes from the 2010 Presidential election for representatives from three major political parties and from 2010 General Election data. All of the data points were obtained through reliable government sources, two of which are, the Department of Elections website of Sri Lanka, and the National News Paper statistics (2010). The study contrasts the distribution of the first digit of election results against the Benford’s Law benchmark. After obtaining the results, we organize the data and find median, mean, mode and standard deviation. The preliminary results showe that the data does not align with Benford’s law predictions. In other words, it shows that the data does not follow the law where the mean is larger than the median and there is a positive skewness then it likely follows a Benford’s distribution. The distribution of the first digit of actual data for three parties disagrees with Benford's law. This misalignment is more pronounced for the winning party than for the second and third place parties, respectively. We, therefore, look forward to run the data through several critical analyses and observing if there shall be any fraud or manipulation in numbers.

scholarly journals Benford’s Law Application: Case of Elections in Sri Lanka

2020 ◽  
Author(s):  
Richmond Sam Quarm ◽  
Mohamed Osman Elamin Busharads
Keyword(s):  
Sri Lanka ◽  
Political Parties ◽  
Presidential Election ◽  
Benford’S Law ◽  
Third Place ◽  
Benford's Law ◽  
Data Points ◽  
The Mean ◽  
Election Fraud ◽  
Election Results

Sri Lanka, like many developing countries has been involved in a circle of allegations of election fraud. Usually these claims are pronounced more by losing parties. This study uses Benford’s law, a law of probability distribution of digits, to investigate whether the election fraud claims might have merit. A sample in this study is made of 808 election data. This data comes from the 2010 Presidential election for representatives from three major political parties and from 2010 General Election data. All of the data points were obtained through reliable government sources, two of which are, the Department of Elections website of Sri Lanka, and the National News Paper statistics (2010). The study contrasts the distribution of the first digit of election results against the Benford’s Law benchmark. After obtaining the results, we organize the data and find median, mean, mode and standard deviation. The preliminary results showe that the data does not align with Benford’s law predictions. In other words, it shows that the data does not follow the law where the mean is larger than the median and there is a positive skewness then it likely follows a Benford’s distribution. The distribution of the first digit of actual data for three parties disagrees with Benford's law. This misalignment is more pronounced for the winning party than for the second and third place parties, respectively. We, therefore, look forward to run the data through several critical analyses and observing if there shall be any fraud or manipulation in numbers.

An Introduction to Benford's Law

2015 ◽  
Author(s):  
Arno Berger ◽  
Theodore P. Hill
Keyword(s):  
Random Variables ◽  
Benford’S Law ◽  
Late Nineteenth Century ◽  
Comprehensive Treatment ◽  
Open Problems ◽  
Significant Digit ◽  
Wide Range ◽  
Benford's Law ◽  
Primary Reference ◽  
Basic Facts

This book provides the first comprehensive treatment of Benford's law, the surprising logarithmic distribution of significant digits discovered in the late nineteenth century. Establishing the mathematical and statistical principles that underpin this intriguing phenomenon, the text combines up-to-date theoretical results with overviews of the law's colorful history, rapidly growing body of empirical evidence, and wide range of applications. The book begins with basic facts about significant digits, Benford functions, sequences, and random variables, including tools from the theory of uniform distribution. After introducing the scale-, base-, and sum-invariance characterizations of the law, the book develops the significant-digit properties of both deterministic and stochastic processes, such as iterations of functions, powers of matrices, differential equations, and products, powers, and mixtures of random variables. Two concluding chapters survey the finitely additive theory and the flourishing applications of Benford's law. Carefully selected diagrams, tables, and close to 150 examples illuminate the main concepts throughout. The book includes many open problems, in addition to dozens of new basic theorems and all the main references. A distinguishing feature is the emphasis on the surprising ubiquity and robustness of the significant-digit law. The book can serve as both a primary reference and a basis for seminars and courses.

Can Vote Countsʼ Digits and Benfordʼs Law Diagnose Elections?

Benford's Law ◽  
2015 ◽  
Author(s):  
Walter R. Mebane,
Keyword(s):  
United States ◽  
Political Parties ◽  
Strategic Behavior ◽  
The United States ◽  
Benford’S Law ◽  
Conditional Mean ◽  
Benford's Law ◽  
Election Fraud

This chapter illustrates how the conditional mean of precinct vote counts' second digits can respond to strategic behavior by voters in response to the presence of a coalition among political parties. The digits in vote counts can help diagnose both the strategies voters use in elections and nonstrategic special mobilizations affecting votes for some candidates. The digits can also sometimes help diagnose some kinds of election fraud. The claim that deviations in vote counts' second digits from the distribution implied by Benford's law is an indicator for election fraud, generally fails for precinct vote counts. This chapter shows that such tests routinely fail in data from elections in the United States, Germany, Canada and Mexico, countries where it is usually thought that there is negligible fraud.

scholarly journals Was there any widespread fraud in 2020 presidential election? What does Benford's Law say?

2021 ◽  
Vol 10 (3) ◽  
Author(s):  
Deeya Datta ◽  
David Banks
Keyword(s):  
Presidential Election ◽  
Forensic Analysis ◽  
Benford’S Law ◽  
Large Set ◽  
Donald Trump ◽  
Comprehensive Investigation ◽  
Electoral Fraud ◽  
Benford's Law ◽  
Election Fraud ◽  
Basic Statistical Analysis

Fair elections free of any interference are integral tenets of any functioning democracy, and widespread election fraud is undoubtedly a serious threat to a free republic. While instances of electoral fraud are much more prevalent in countries with illiberal democracies, the U.S has recently faced such an accusation. Although he was unable to provide any concrete evidence, the former U.S. President Donald Trump accused his opponent, Joe Biden, now president, of electoral fraud after the presidential election. Fortunately, election forensics are often successful in investigating the validity of such fraud allegations. In this paper, I applied Benford’s law, a rule that should stand up to any large set of natural numbers, such as un-tampered electoral data. Using this law and basic statistical analysis of votes of U.S. counties for candidates of the two major parties, I completed a forensic analysis to investigate Mr. Trump’s allegation. My comprehensive investigation does not find any evidence supporting his allegation.

Random-Number Guessing and the First Digit Phenomenon

1988 ◽  
Vol 62 (3) ◽  
pp. 967-971 ◽  
Author(s):  
Theodore P. Hill
Keyword(s):  
Random Number ◽  
Benford’S Law ◽  
Empirical Observation ◽  
Significant Digit ◽  
Physical Constants ◽  
Benford's Law ◽  
Empirical Regularities ◽  
Newspaper Articles

To what extent do individuals “absorb” the empirical regularities of their environment and reflect them in behavior? A widely-accepted empirical observation called the First Digit Phenomenon or Benford's Law says that in collections of miscellaneous tables of data (such as physical constants, almanacs, newspaper articles, etc.), the first significant digit is much more likely to be a low number than a high number. In this study, an analysis of the frequencies of the first and second digits of “random” six-digit numbers guessed by people suggests that people's responses share some of the properties of Benford's Law: first digit 1 occurs much more frequently than expected; first digit 8 or 9 occurs much less frequently; and the second digits are much more uniformly distributed than the first.

scholarly journals Benford’s law in the Gaia universe

2020 ◽  
Vol 642 ◽  
pp. A205
Author(s):  
Jurjen de Jong ◽  
Jos de Bruijne ◽  
Joris De Ridder
Keyword(s):  
Bayesian Approach ◽  
Dynamic Range ◽  
Zero Point ◽  
Atomic Data ◽  
Benford’S Law ◽  
Selection Function ◽  
Significant Digit ◽  
Benford's Law ◽  
The Difference ◽  
Digit Frequencies

Context. Benford’s law states that for scale- and base-invariant data sets covering a wide dynamic range, the distribution of the first significant digit is biased towards low values. This has been shown to be true for wildly different datasets, including financial, geographical, and atomic data. In astronomy, earlier work showed that Benford’s law also holds for distances estimated as the inverse of parallaxes from the ESA HIPPARCOS mission. Aims. We investigate whether Benford’s law still holds for the 1.3 billion parallaxes contained in the second data release of Gaia (Gaia DR2). In contrast to previous work, we also include negative parallaxes. We examine whether distance estimates computed using a Bayesian approach instead of parallax inversion still follow Benford’s law. Lastly, we investigate the use of Benford’s law as a validation tool for the zero-point of the Gaia parallaxes. Methods. We computed histograms of the observed most significant digit of the parallaxes and distances, and compared them with the predicted values from Benford’s law, as well as with theoretically expected histograms. The latter were derived from a simulated Gaia catalogue based on the Besançon galaxy model. Results. The observed parallaxes in Gaia DR2 indeed follow Benford’s law. Distances computed with the Bayesian approach of Bailer-Jones et al. (2018, AJ, 156, 58) no longer follow Benford’s law, although low-value ciphers are still favoured for the most significant digit. The prior that is used has a significant effect on the digit distribution. Using the simulated Gaia universe model snapshot, we demonstrate that the true distances underlying the Gaia catalogue are not expected to follow Benford’s law, essentially because the interplay between the luminosity function of the Milky Way and the mission selection function results in a bi-modal distance distribution, corresponding to nearby dwarfs in the Galactic disc and distant giants in the Galactic bulge. In conclusion, Gaia DR2 parallaxes only follow Benford’s Law as a result of observational errors. Finally, we show that a zero-point offset of the parallaxes derived by optimising the fit between the observed most-significant digit frequencies and Benford’s law leads to a value that is inconsistent with the value that is derived from quasars. The underlying reason is that such a fit primarily corrects for the difference in the number of positive and negative parallaxes, and can thus not be used to obtain a reliable zero-point.

When Does the Second-Digit Benford’s Law-Test Signal an Election Fraud?

2011 ◽  
Vol 231 (5-6) ◽  
Author(s):  
Susumu Shikano ◽  
Verena Mack
Keyword(s):  
Statistical Method ◽  
Specific Form ◽  
Benford’S Law ◽  
Parliamentary Election ◽  
Test Results ◽  
Electoral Fraud ◽  
Wide Range ◽  
Benford's Law ◽  
Election Fraud ◽  
Minimal Information

SummaryDetecting election fraud with a simple statistical method and minimal information makes the application of Benford’s Law quite promising for a wide range of researchers. Whilst its specific form, the Second-Digit Benford’s Law (2BL)-test, is increasingly applied to fraud suspected elections, concerns about the validity of its test results have been raised. One important caveat of this kind of research is that the 2BL-test has been applied mostly to fraud suspected elections. Therefore, this article will apply the test to the 2009 German Federal Parliamentary Election against which no serious allegation of fraud has been raised. Surprisingly, the test results indicate that there should be electoral fraud in a number of constituencies. These counter intuitive results might be due to the naive application of the 2BL-test which is based on the conventional χ

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