Benford’s Law for Telemetry Data of Wildlife

Benford’s law (BL) specifies the expected digit distributions of data in social sciences, such as demographic or financial data. We focused on the first-digit distribution and hypothesized that it would apply to data on locations of animals freely moving in a natural habitat. We believe that animal movement in natural habitats may differ with respect to BL from movement in more restricted areas (e.g., game preserve). To verify the BL-hypothesis for natural habitats, during 2015–2018, we collected telemetry data of twenty individuals of wild red deer from an alpine region of Austria. For each animal, we recorded the distances between successive position records. Collecting these data for each animal in weekly logbooks resulted in 1132 samples of size 65 on average. The weekly logbook data displayed a BL-like distribution of the leading digits. However, the data did not follow BL perfectly; for 9% (99) of the 1132 weekly logbooks, the chi-square test refuted the BL-hypothesis. A Monte Carlo simulation confirmed that this deviation from BL could not be explained by spurious tests, where a deviation from BL occurred by chance.

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Self-reported income data: are people telling the truth?

Journal of Financial Crime ◽

10.1108/jfc-08-2019-0113 ◽

2020 ◽

Vol 27 (4) ◽

pp. 1349-1359

Author(s):

Fernando Antonio Ignacio González

Keyword(s):

Household Survey ◽

Executive Branch ◽

Benford’S Law ◽

Mean Deviation ◽

Public Officials ◽

Chi Square ◽

Content Type ◽

Anomalous Data ◽

Benford's Law ◽

Chi Square Test

Purpose This paper aims to detect anomalous data in income reports of Argentina, including personal income – from a sample of households – and statements of public officials. Design/methodology/approach A widely known technique in forensic accounting – such as Benford’s Law – is used. The Chi-square test and the absolute mean deviation are used for verification. The databases consulted include the income declared by households in the Permanent Household Survey – for the 2003-2017 period – and the capital declarations of high-ranking public officials – for the period 1999-2017. Findings The results suggest that income reported in the Encuesta Permanente de Hogares do not follow a Benford´s distribution, and the degree of conformity with this decreases significantly between 2007 and 2015 – coincident with the intervention period of the Instituto Nacional de Estadísticas y Censos. Patrimonial statements of public officials present an acceptable level of compliance with Benford’s law, especially among those of the legislative branch (in more than 90% of cases) although to a lesser extent among officials of the executive branch. Practical implications The results suggest that income reports from the Permanent Household Survey, for the period 2007-2015, should be used with reservations because of their possible manipulation. Originality/value During the intervention of the official statistics institute in Argentina (2007-2015), the idea of lack of credibility of its reports has been disseminated. To date, however, there is no empirical evidence to support it related to income.

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The empirical analysis of financial reports of companies in Croatia: Benford distribution curve as a benchmark for first digits

Croatian Review of Economic Business and Social Statistics ◽

10.2478/crebss-2019-0014 ◽

2019 ◽

Vol 5 (2) ◽

pp. 90-100

Author(s):

Ivana Cunjak Mataković

Keyword(s):

Distribution Curve ◽

Stock Exchange ◽

Financial Statements ◽

Benford’S Law ◽

Chi Square ◽

Financial Reports ◽

Benford's Law ◽

Chi Square Test ◽

Kolmogorov Smirnov ◽

The Empirical Analysis

AbstractThe financial numbers game is unfortunately alive and doing well. One of the forensic accounting techniques is based on Benford’s Law and is used for the detection of unusual transactions, anomalies or trends. The aim of this paper is to test whether the financial statements of Croatian companies deviate from Benford’s Law distribution. The financial statements of 24 companies that are in the pre-bankruptcy settlement process and 24 companies that are not in the pre-bankruptcy settlement process were analysed using the Benford’s Law test of the first digit distribution for the period from 2015 to 2018. The data used to calculate the first digits of distribution were taken from the Zagreb Stock Exchange. The chi-square test has shown that the observed companies that are not in the process of pre-bankruptcy settlement do not have the first digit distribution which follows the Benford’s Law distribution. The Kolmogorov-Smirnov Z test has shown that the distribution of the first digits from the financial statements of companies listed on the Zagreb Stock Exchange fits to Benford’s Law distribution.

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

Stats ◽

10.3390/stats4020027 ◽

2021 ◽

Vol 4 (2) ◽

pp. 419-453

Author(s):

Alex Ely Kossovsky

Keyword(s):

Research Question ◽

Real Life ◽

Benford’S Law ◽

Data Sets ◽

Chi Square ◽

Data Set ◽

Life Data ◽

Real Life Data ◽

Benford's Law ◽

Chi Square Test

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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On the possibility of Benfordʼs Law application in accounting fraud detection

Zeszyty Teoretyczne Rachunkowości ◽

10.5604/01.3001.0014.5549 ◽

2020 ◽

Vol 110 (166) ◽

pp. 11-30

Author(s):

Mateusz Baryła

Keyword(s):

Research Method ◽

Goodness Of Fit ◽

Human Mind ◽

Benford’S Law ◽

Accounting Fraud ◽

Process Data ◽

Goodness Of Fit Test ◽

Chi Square ◽

Polish Literature ◽

Benford's Law

Purpose: The purpose of the article is to indicate that, theoretically and practically, Benford’s Law can be applied in order to detect accounting frauds. Methodology/approach: The paper provides an overview of current regulations and experts’ opinions published in the existing literature and internet sources. Moreover, data analysis was used as a research method. Findings: The results of assessing the conformity of the first two significant digits of distribution of foreign revenues from the sales of finished products to Benford’s Law (using the chi-square goodness of fit test) showed that in the case of a proper accounting process, one cannot reject the hypothesis that the data conform to Benford’s Law. On the other hand, the analysis of intentionally falsified foreign revenues led to the conclusion that in the case of an improper accounting process, data, in general, does not conform to Benford’s Law. Research limitations/implications: In the study, it was assumed that the human mind generates false val-ues of accounting entries, and the number of attempts to commit fraud was limited to 10. Originality/value: The article extends the existing knowledge of using Benfordʼs Law in detecting ac-counting fraud in the Polish literature.

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Pesticides and the evolution of the genetic structure of Anopheles coluzzii populations in some localities in Benin (West Africa)

Malaria Journal ◽

10.1186/s12936-019-3036-z ◽

2019 ◽

Vol 18 (1) ◽

Cited By ~ 1

Author(s):

Arsène Jacques Y. H. Fassinou ◽

Come Z. Koukpo ◽

Razaki A. Ossè ◽

Fiacre R. Agossa ◽

Roseric Azondékon ◽

...

Keyword(s):

Genetic Structure ◽

Genetic Differentiation ◽

Anopheles Coluzzii ◽

Natural Habitats ◽

Chi Square ◽

Agroecological Zones ◽

Chi Square Test ◽

Hardy Weinberg Equilibrium ◽

Study Sites ◽

Homozygous Resistant

Abstract Background Changes in the natural habitats of insect groups are determined the genetic polymorphisms between individuals. The objective of this study was to establish the genetic structure of the Anopheles coluzzii populations in four localities of Benin. Methods Insecticide surveys and larval sampling were conducted on 4 study localities, including Cotonou, Ketou, Zagnanado, and Sô-Ava. Molecular characterizations were performed on the Anopheles mosquitoes collected with the allelic and genotypic frequencies of kdr gene determined. The multiple comparison Chi square test for proportions was performed with R version 3.3.3. Next, the observed heterozygosity, expected heterozygosity, and indices of fixation, and genetic differentiation were estimated. Finally, the Hardy–Weinberg equilibrium (EHW) was determined to assess whether panmixia exists in the different populations of mosquitoes of the agroecological zones under study. Results Carbamates, pyrethroids, organophosphorus and organochlorines use have been reported in all localities except Sô-Ava. Anopheles coluzzii was strongly represented across all study localities. The L1014F allele was observed in the localities of Kétou, Cotonou and Zagnanado. Likewise, insecticide selection pressure of homozygous resistant individuals (L1014F/L1014F) was significantly higher in Kétou, Cotonou and Zagnanado (p value < 0.05). Surprisingly in Sô-Ava, a relatively high frequency of the L1014F allele despite the reported absence of pesticide use was observed. All mosquito populations were found to be deficient in heterozygosity across the study sites (FIS< 0). No genetic differentiation (FST< 0) was observed in the localities of Zagnanado and Kétou. Conclusion The survey on the use of insecticides showed that insecticide selection pressures differ across the investigated localities. It would be desirable to rotate or apply formulations of combined products with different modes of action. Doing so would enable a better management of resistant homozygous individuals, and mitigate the resistance effect of commonly used insecticides.

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Testing of Published Information on Greenhouse Gas Emissions. Conformity Analysis with the Benford’s Law Method

Audit Financiar ◽

10.20869/auditf/2020/160/029 ◽

2020 ◽

Vol 18 (160) ◽

pp. 821-830

Author(s):

Alexandru-Teodor CORACIONI ◽

◽

Tatiana DANESCU ◽

Keyword(s):

Greenhouse Gas Emissions ◽

Greenhouse Gas ◽

Benford’S Law ◽

Published Data ◽

Practical Case ◽

Chi Square ◽

Gas Emissions ◽

Benford's Law ◽

Financial Audit ◽

Analytical Review

The issue of greenhouse gas emissions and climate change cannot be overlooked from the scope of concern essential to develop the financial audit profession and implicitly the activity of the financial auditor. Within its professional field, the analysis of data quality on these issues requires applying divers analytical review methods, which incorporates unique statistical or mathematical rules. One of these is the analytical review procedure based on testing compliance of data distribution with Benford’s Law. We present a practical case for testing the verisimilitude of data represented by greenhouse gas emissions, based on the Eurostat database. We applied the Benford's Law method for the first four digits, and the results were tested for likelihood by statistical methods such as Chi-square test or the Kolmogorov-Smirnov test. Information on greenhouse gas emissions is the basis for specific environmental policy decisions, which can be considered at the microor macroeconomic level. If this information is affected by subjective influences, then economic decisions or environmental policies will also be affected. Therefore, the objective of this research to test the plausibility of published data on greenhouse gas emissions proves its usefulness in relation with the actions performed by the economic and social players towards a sustainable development of the economy.

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THE APPLICATION OF BENFORD'S LAW IN PSYCHOLOGICAL PRICING DETECTION

Zbornik radova - Journal of Economy and Business ◽

10.46458/27121097.2018.24.37 ◽

2018 ◽

pp. 37-57

Keyword(s):

Uniform Distribution ◽

Benford’S Law ◽

Mean Absolute Deviation ◽

Chi Square ◽

Absolute Deviation ◽

Discrete Uniform Distribution ◽

Benford's Law ◽

Kolmogorov Smirnov ◽

Discrete Uniform ◽

Chi Square Tests

This paper presents the application of Benford's law in psychological pricing detection. Benford's law is naturally occurring law which states that digits have predictable frequencies of appearance with digit one having the highest frequency. Psychological pricing is one of the marketing pricing strategies directed on price setting which have the psychological impact on certain consumers. In order to investigate the application of Benford's law in psychological pricing detection, Benford's law is observed in the case of first and last digits. In order to inspect if the first and last digits of the observed prices are distributed according to the Benford’s law distribution or discrete uniform distribution respectively, mean absolute deviation measure, chi-square tests and Kolmogorov-Smirnov Z tests are used. Results of the analysis conducted on three price datasets have shown that the most dominating first digits are 1 and 2. On the other side, the most dominating last digits are 0, 5 and 9 respectively. The chi-square tests and Kolmogorov-Smirnov Z tests have showed that, at significance level of 5%, none of the three observed price datasets does have first digit distribution that fits to the Benford’s law distribution. Likewise, mean absolute deviation values have shown that there are large differences between the last digit distributions and the discrete uniform distribution implying psychological pricing in all price datasets.

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A multivariate test for detecting fraud based on Benford’s law, with application to music streaming data

Statistical Methods & Applications ◽

10.1007/s10260-021-00582-6 ◽

2021 ◽

Author(s):

Nermina Mumic ◽

Peter Filzmoser

Keyword(s):

Test Performance ◽

Compositional Data ◽

Streaming Data ◽

Benford’S Law ◽

Chi Square ◽

Multivariate Test ◽

Relative Information ◽

Benford's Law ◽

Kolmogorov Smirnov ◽

Music Streaming

AbstractBenford’s law became a prevalent concept for fraud and anomaly detection. It examines the frequencies of the leading digits of numbers in a collection of data and states that the leading digit is most often 1, with diminishing frequencies up to 9. In this paper we propose a multivariate approach to test whether the observed frequencies follow the theoretical Benford distribution. Our approach is based on the concept of compositional data, which examines the relative information between the frequencies of the leading digits. As a result, we introduce a multivariate test for Benford distribution. In simulation studies and examples we compare the multivariate test performance to the conventional chi-square and Kolmogorov-Smirnov test, where the multivariate test turns out to be more sensitive in many cases. A diagnostics plot based on relative information allows to reveal and interpret the possible deviations from the Benford distribution.

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The Full Database of Countries with Potential COVID-19 Data Misreport based on Benford’s Law

10.1101/2020.12.04.20243832 ◽

2020 ◽

Author(s):

Ahmad Kilani ◽

Georgios P. Georgiou

Keyword(s):

Benford’S Law ◽

Mean Absolute Deviation ◽

Chi Square ◽

Absolute Deviation ◽

The World ◽

Benford's Law

AbstractThe aim of this database is to provide researchers and scholars a unified database for potential data misreport by 171 countries regarding their COVID-19 daily reported cases. The analysis employs three different tests (chi-square, Kuiper, and Mean Absolute Deviation) to determine if the data given by each country in the world fit Benford’s Law.

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Hubungan Antara Menyusui Dengan Involusi Uterus Pada Ibu Nifas Fisiologis Di RSIA Aura Syifa Kabupaten Kediri

Jurnal Ilmu Kesehatan ◽

10.32831/jik.v5i2.136 ◽

1970 ◽

Vol 5 (2) ◽

pp. 64

Author(s):

Eny Sendra ◽

Dewi Indriani

Keyword(s):

Breast Feeding ◽

Sampling Technique ◽

Chi Square ◽

Cross Sectional ◽

Fisher Test ◽

Key Word ◽

Chi Square Test ◽

The Moment

Breast feeding is giving milk to be drunk to the baby from the breast. Uterus involution is a process how the uterus return to the condition back, before pregnanting after bearing. At the moment of suckling, happens a stimulus and brings the hormones out, such as oksitosin uses not only to stimulate some muscles constraction but also to stimulate the uterus, so that the process of uterus involution happens foster. According to the explanations above, the research aimed to know about the correlation between breast feeding and uterus involution. This research’s design was, cross sectional by the population of all childbirth mothers approximately 50 persons / month. By using accidental sampling technique we got 21 sample respondents. The place of research in RSIA Aura Syifa in Kediri Regency on 16th until 22nd of June 2009. From this research’s result, we got 14 persons (66,67%) with normal uterus involution, suckled in a good way, one person (4,67%) with normal uterus involution, suckled in a wrong way, 2 persons (9,52%) with abnormal uterus involution, suckled in a good way and 4 persons (19,05%) with abnormal uterus involution, suckled in a wrong way. Statistic test which used chi-square test, counted the probability frequency in advance, from that we got 3 columns with the score, less than 5, so that chi-square can not be continued and by doing exact fisher test, the score was 0,001. Because P with the grade mistake 0,05 smaller, so the conclusion was “Ho” is rejected, it meant “there was correlation between suckling and uterus involution”. Key Word : Breast feeding, uterus involution

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