Law of the leading digits and the ideological struggle for numbers
aa r X i v : . [ phy s i c s . d a t a - a n ] N ov October 8, 2018 21:22 arxiv
The law of the leading digits and the world religions
T. A. Mir ∗ Nuclear Research Laboratory, Astrophysical Sciences Division,Bhabha Atomic Research Centre,Srinagar-190 006, Jammu and Kashmir, India ∗ [email protected]
1. Abstract
Benford’s law states that the occurrence of significant digits in many data sets is notuniform but tends to follow a logarithmic distribution such that the smaller digitsappear as first significant digits more frequently than the larger ones. We investi-gate here numerical data on the country-wise adherent distribution of seven majorworld religions i.e. Christianity, Islam, Buddhism, Hinduism, Sikhism, Judaism andBaha’ism to see if the proportion of the leading digits occurring in the distributionconforms to Benford’s law. We find that the adherent data of all the religions, ex-cept Christianity, excellently does conform to Benford’s law. Furthermore, unlikethe adherent data on Christianity, the significant digit distribution of the three ma-jor Christian denominations i.e. Catholicism, Protestantism and Orthodoxy obeysthe law. Thus in spite of their complexity general laws can be established for theevolution of the religious groups.
2. Keywords
Benford’s law; religion; adherents
3. Introduction
In our daily life, we come across huge sets of numerical data being generated due todifferent activities both human as well as natural. What could be common betweentwo seemingly unrelated data sets like the size of files stored on a computer and thegeographical area of the countries? An intuitive reply would be nothing. However,the occurrence of significant digits in many data sets is governed by a commonlogarithmic distribution which is defined by the equation called Benford’s law [1].The first significant digit of a number is the first non-zero digit on its extreme leftlike 5 for 521 and 1 for 0.0184 respectively. The establishment of Benford’s law was1ctober 8, 2018 21:22 arxiv based on the curious observation made by Simon Newcomb who observed and saidthat the initial pages of the logarithmic tables were more worn out than the laterones due to the fact that people look more often for numbers with smaller first non-zero digits [2]. The observation was popularized by Frank Benford (hence the nameBenford’s law) who then analyzed a large number of unrelated data sets collectedfrom diverse fields e.g. the physical constants, the atomic and molecular masses, theareas of countries, the length of rivers etc., and concluded that the occurrence offirst significant digits in all his data sets does follow a logarithmic distribution andhence established the law in the form of following mathematical equation [1]. P ( d ) = log (1 + 1 d ) , d = 1 , , ..., =11% foreach digit from 1 to 9 if all are equally likely to occur as the first significant digit.Despite many of its properties like base and scale invariance being unraveled [3, 4]over more than a century after its discovery a complete explanation of Benford’s lawhas remained elusive [5]. However, increasing number of data sets from unrelatedphenomena have been tested to conform to this law with high statistical accuracy.After remaining obscure for decades, the law is finding wide spread applications. Ineconomics, it has been validated for stock market prices [6] and applied in detect-ing the fraudulent data submitted by the companies [7], verifying the reliability ofmacroeconomic data of the countries [8] and in the determination of winning bidsin e-Bay auctions [9]. In physics the law has been successfully applied to numericaldata on physical constants [10], atomic spectra [11], decay width of hadrons [12],magnitude and depth of earthquakes [6,13] and mantissa distribution of pulsars [14].The law has been utilized in biological sciences to check the veracity of the data onclinical trials and discovery of drugs [15, 16], study of diseases and genes [17]. Thelaw is used in optimizing the size of computer files [18], enhancing the computingspeed [19] and appearance of numbers on the internet [20]. On the social sciencesfront the law has been used to detect election frauds and anomalies [21].We show here the applicability of Benford’s law to the field of religious demography.Though very important, the research on religious demography is still in infancy andadmittedly inconsistencies in world religion adherent data cannot be ruled out com-pletely. Besides reshaping the social landscape, the change in religious demographicsis also affecting the political decisions of various countries. To prevent problems likemaking a flawed policy decision due to the use of some erroneous data it is theobligation of the researchers to reassure the policy makers about the quality ofctober 8, 2018 21:22 arxiv the data. We analyze the available data on the country-wise adherent distributionof seven major world religions Christianity, Islam, Buddhism, Hinduism, Sikhism,Judaism and Baha’ism. We find that country-wise adherent data on all religionsexcept Christianity follow Benford’s law. We also reveal the validity of Benford’slaw on Catholicism, Protestantism and Orthodoxy, the three major denominationsof Christianity.
4. Data
Due to the developmental stage of the religious demography research, obtainingcomplete data on adherents of a particular faith and its distribution from a singlesource is not quite easy. The Wikipedia combines the available data from differentsources like the CIA World Factbook, US State Department’s International Reli-gious Freedom Report, Pew Research Center, Adherents.com, Joshua Project etc.to arrive at comprehensive data tables for the country-wise adherent distribution ofmajor world religions. The data on country-wise number of Muslims are taken fromPew Research Center study on the size and distribution of world’s Muslim popula-tion [22, 23]. The number of adherents of Baha’i faith in each country is based onthe 2005 estimates of World Christian Database as reported in The Association ofReligion Data Archive (ARDA) [24].
Data analysis and Results
The detailed statistical analysis of the country-wise adherent distribution is shownin Tables 1 and 2. The N Obs , the number of counts appearing in the correspondingdata set as first significant digit, against each digit from 1 to 9 are shown foreach religion in columns 2, 3, 4, 5, 6 of Tables 1 and 2. We also show N Ben , thecorresponding counts (in brackets) for each digit as predicted by Benford’s law: N Ben = N log (1 + 1 d ) (2)along with the root mean square error (∆ N ) calculated from the binomial distribu-tion [12] ∆ N = p N P ( d )(1 − P ( d )) (3)where N for each religion is the total number of countries its adherents are reportedto exist. For example, as shown in column 2 of Table 1, the adherents of Christianityare reported in 205 territories [25]. The observed count for digit 1 as first significantdigit is 50 whereas the expected count from Benford’s law is 61.7 with an error ofabout 6.6. To evaluate the degree of agreement between the observed and expectedfirst digit distributions let us first write the Null Hypothesis , H O that the observeddistribution of the first significant digit in each of the case we consider is same asexpected on the basis of Benford’s law. To test the null hypothesis we carry out thectober 8, 2018 21:22 arxiv Pearson’s χ test. χ ( n −
1) = n X i =1 ( N Obs − N Ben ) N Ben (4)In our case n = 9 which means we have n − χ (8)=15.507 which is the critical value for acceptance orrejection of null hypothesis. If the value of the calculated χ is less than the criticalvalue then we accept the null hypothesis and conclude that the data fits Benford’slaw.Christianity has the largest number of adherents spread across 205 countries (col-umn 2 of Table 1). The χ = 16.419 (the last row and column 2 of Table 1) is largerthan χ (8) = 15 .
507 and thus forcing us to reject the null hypothesis indicating inturn that the data on the country-wise adherent distribution of Christianity doesnot follow Benford’s law. We further checked the behavior of three largest Christiandenominations, Catholicism (197 countries) [26], Protestantism (171 countries) [27]and Orthodoxy (42 countries) [28] as shown in Columns 3, 4 and 5. Surprisingly theirrespective χ values 10.143, 5.208, 6.946 turn out to be lower than χ (8) = 15 . χ = 10 .
646 is well below the critical value of 15 .
507 andnull hypothesis is accepted. In Table 2 we show the data on Baha’ism (175 coun-tries) [24], Buddhism (129 countries) [29], Hinduism (97 countries) [30], Sikhism (47countries) [31] and Judaism (107 countries) [32]. The smaller values of the Pear-son’s χ make the null hypothesis acceptable and hence Benford’s law describesthe distribution of significant digits in each case. The Wikipedia table on the num-ber of adherents of Buddhism shows two values for some countries. Accordingly weconsider both the situations taking into account the lower and the upper valuesseparately. When smaller values are counted for the Buddhism (not shown here),the Pearson’s χ .
633 turns out to be smaller than the critical value. The lowestvalue of χ .
786 occurs for Buddhism when only the larger values are considered forcountries where two values are given. Further, although the sample size of the ad-herent data on Sikhism is quite small, reported only in 47 countries, the χ = 6 . χ =7.310 being smaller than the criticalvalue 15.507 shows that Benford’s law describes the adherent data of major worldreligions with a reasonable accuracy. The results reported in the tables can readilybe appreciated from Figs. 1-3. It can be seen that the occurrence of the signifi-cant digits in country-wise adherent distribution of religions closely resembles thosepredicted by Benford’s law.ctober 8, 2018 21:22 arxiv Table 1: The significant digit distribution of country-wise Christian, Catholic,Protestant, Orthodox and Muslim populations
First Digit Christian Catholic Protestant Orthodox Muslim(205) (197) (171) (42) (184)1 50 (61.7 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Pearson χ Table 2: The significant digit distribution of country-wise Baha’i, Buddhist, Hindu,Sikh and Jewish populations
First Digit Baha’i Buddhist Hindu Sikh Jew(175) (129) (97) (47) (107)1 61 (52.9 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Pearson χ Table 3: The significant digit distribution of country-wise adherents of all religions
First Digit 1 2 3 4 5 6 7 8 9 Total N Obs
289 169 118 84 66 69 64 36 49 N Ben
Pearson χ In the case of any major deviations from the predictions of Benford’s law, thedistributions as shown in figures could hardly be revealed. Particularly from Fig.ctober 8, 2018 21:22 arxiv Fig. 1: Observed and Benford distributions of significant digits for country-wiseadherents of Christianity, its three major denominations i.e. Catholicism, Protes-tantism and Orthodoxy, Islam and Baha’ismctober 8, 2018 21:22 arxiv Fig. 2: Observed and Benford distributions of significant digits for country-wiseadherents of Buddhism, Hinduism, Judaism and Sikhism3 it is clear that agreement of the observed proportion of occurrence of significantdigits for all the religions to those predicted by Benford’s law is excellent.
5. Discussion
Despite being a key social attribute and the largest scale phenomenon involvingpeople beyond the geographical boundaries of countries, the adherence to a reli-gion, has received relatively lesser scientific attention [33]. Furthermore, religionbeing a dominant force in society for both cohesion and conflict [34], when it comesto the scientific study of religions the most debated question, having serious socialand political consequences, is the estimation of size distribution of different reli-gious groups [23, 35, 36]. Studies have shown that the growth dynamics of religiousactivities are similar to those found in economic and scientific activities [37]. Thectober 8, 2018 21:22 arxiv Fig. 3: Observed and Benford distributions of significant digits for country-wiseadherents of all religionsadherent distribution of major world religions has been shown to obey Zipf’s andPareto distributions and their evolution has been described using the same equa-tions as for the crystallization process [38, 39]. A recent mathematical model haspredicted the extinction of religion in societies which perceive the utility of not ad-hering to be greater than the utility of adhering [40].Here we have shown that populations of major world religions follow Benford’s law.It is worthwhile to mention here that F. Benford used population size of U.S. townsas input data to test and examine the accuracy of his observation [1]. Later on, thepopulation of the 198 countries of world have been shown to follow the first digitlaw [41]. However, being undertaken by the individual governments, the census esti-mates of the total populations of countries are expected to be reliable. Same is nottrue for the estimates of adherents of different religions. Several religions competewith each other to increase the number of their adherents. Such followers can eitherbe converted from one religion to another or become followers of a particular reli-gion though previously being associated with none [42]. This continuous evolutionof religions puts serious restraints on determining the exact number of adherents ofa particular religion, a task which is further complicated due to government favorto state religions in certain countries, government restriction on religious census orbiased data from different religious associations for political propaganda [43]. Thuserrors are expected to be found in the data on religious demographics and datamust be subjected to scientic scrutiny for any alterations.Benford’s law holds for those data sets in which the occurrence of numbers is freefrom any restrictions. It has been found that tampered, unrelated or fabricatednumbers usually do not follow Benford’s law [44]. Significant deviations from thectober 8, 2018 21:22 arxiv Benford distribution may indicate fraudulent or corrupted data [45]. In the light ofthese issues it would be interesting to check if the adherent data of major worldreligions submit to Benford’s law and any departure from the law may alert theresearcher about possible data misrepresentation.In our study we observed that the adherent data of three major Christian denomi-nations follow Benford’s law. However, when Christianity is considered as a singlereligious group, the distribution of the significant digits of the adherent data devi-ates from the predictions of Benford’s law. The deviation could possibly arise fromthe incorrect method used or erroneous grouping of data from different sources inWikipedia. Further, the deviation from the Benford distribution may also arise fromoverestimation/underestimation of the number of Christian adherents in some coun-tries. The absence of a single religion database has forced many people includingscholars to turn to Wikipedia for information on major world religions. But rele-vant Wikipedia data has been assumed to be devoid of sufficient academic rigor andconsistency [23]. However, the prevalence of Benford’s law in the adherent data onthree largest Christian denominations, Buddhism, Hinduism, Sikhism, Judaism andBaha’ism infuses some confidence in the quality of the data available on Wikipediaand the method employed to collect and combine the data from different sources toarrive at these comprehensive adherent data tables. Now in the last decade, Muslimsocieties have generated considerable public interest and their population has comeunder increased scrutiny [22, 23]. There are varying estimates of the size of world’sMuslim population. We analyzed the estimates of the global Muslim populationreported in the most recent study done by the Pew Research Center and founda reasonable agreement to Benford’s law which in turn indicates the reliability ofthis demographic survey. The validity of Benford’s law for adherent distributionof major world religions is a further hint that universal growth mechanisms mightexist which can give rise to general laws independent of the particular details of thesystems [37].
6. Conclusion
The religion is the largest scale phenomenon in a society that has a bearing onsocial, economical and political behavior of its adherents. We applied, for the firsttime, Benford’s law to the data on the adherent distribution of seven major worldreligions. We found that the adherent data of Islam, Hinduism, Buddhism, Sikhism,Judaism and Baha’ism follow Benford’s law. Similarly, the adherent data of thethree major Christian denominations Catholicism, Protestantism and Orthodoxyalso follow the law but contrarily cumulative adherent data of Christianity whenconsidered as a single religious group does not. Notwithstanding the complex pathtaken by a religious group to attain a certain state, its macroscopic behavior satisfiesthe simple law of leading digits.ctober 8, 2018 21:22 arxiv Acknowledgments
Suggestions from M. Ausloos and P. M. Ishtiaq are gratefully acknowledged.
References
1. F. Benford, The law of anomalous numbers, Proc. Am. Phil. Soc. 78 (1938)551-572.2. S. Newcomb, Note on the frequency of use of different digits in natural numbers,Am. J. Math. 4 (1881) 39-40.3. T. P. Hill, Base-invariance implies Benford’s law, Proc. Am. Math. Soc. 123 (3)(1995) 887-895.4. T. P. Hill, A statistical derivation of the significant-digit law, Stat. Sci. 10 (4)(1995) 354-363.5. A. Berger, T. P. Hill, Benford’s law strikes back: no simple explanation in sightfor mathematical gem, The Mathematical Intelligencer, 33 (1) (2011) 85-91.6. L. Pietronero et al., Explaining the uneven distribution of numbers in nature:the laws of Benford and Zipf, Physica A 293 (2001) 297-304.7. M. J. Nijrini, Taxpayer compliance application of Benford’s law, The J. Am.Tax. Assoc. 18 (1) (1996) 72-92.8. J. Nye, C. Moul, The political economy of numbers: on the application of Ben-ford’s law to international macroeconomic statistics, B. E. J. Macroeconomics,7 (1) (2007) article 179. D. E. Giles, Benford’s law and naturally occurring prices in certain eBay auc-tions, App. Econ. Lett. 14 (3) (2007) 157-161.10. J. Burke, E. Kincanon, Benford law and physical constants - The distributionof initial Digits, Am. J. Phys. 59 (10) (1991) 952.11. J. C. Pain, Benfords law and complex atomic spectra, Phys. Rev. E. 77 (2008)012102.12. L. Shao, B. Q. Ma, First digit distribution of hadron full width, Mod. Phys.Lett. A 24 (2009) 3275-3282.13. M. Sambridge et al., Benford’s law in the natural sciences, Geo. Phys. Res.Lett. A 37 (2010) L22301.14. L. Shao, B. Q. Ma, Empirical mantissa distributions of pulsars, Astrop. Phys.33 (2010) 255-262.15. T. W. Beer, Terminal digit preference: beware of Benfords law, J. Clin. Pathol.62 (2009) 192.16. M. Orita et al., Use of Benford’s law in drug discovery data, Drug DiscoveryToday, 15 (2010) 328-331.17. J. L. Hernandez Caceres et al., First digit distribution in some biological datasets. Possible explanations for departures from Benford’s Law, El. J. Biomed. 1(2008) 27-35.18. J. L. Barlow, E. H. Bareiss, On roundoff error distributions in floating pointand logarithmic arithmetic, Computing, 34 (4) (1985) 325-347.ctober 8, 2018 21:22 arxiv
19. P. Schatte, On mantissa distributions in computing and Benford’s law, J. Inf.Proc. Cyber. 24 (9) (1988) 443-455.20. S. N. Dorogovtsev et al., Frequency of occurrence of numbers in the WorldWide Web, Physica A. 360 (2006) 548-556.21. W. R. Mebane, Jr., The wrong man is president! Overvotes in the 2000 presi-dential election in Florida, Presp. on Polit. 2 (2004) 525-535.22. Mapping the global muslim population: A report on the size and dis-tribution of the world’s muslim population, Pew Research Center (2009),http://pewforum.org/docs/?DocID=45023. B. J. Grim, B. Hsu, Estimating the global muslim population: size and distri-bution of the world’s muslim population, Interd. J. Res. Relig. 7 (2011) article2.24. Most Baha’i Nations (2005), accessed on 22 Feb 2011, available at http : − c.asp
25. Christianity by country, accessed on 17 Feb 2011, available at http : //en.wikipedia.org/wiki/Christianity − by − country
26. Catholic Church by country, accessed on 21 Feb 2011,available at http : //en.wikipedia.org/wiki/Catholic − by − country ,27. Protestantism by country, accessed on 19 Feb 2011,available at http : //en.wikipedia.org/wiki/P rotestantism − by − country
28. Orthodoxy by country, accessed on 19 Feb 2011,available at http : //en.wikipedia.org/wiki/Orthodoxy − by − country
29. Buddhism by country, accessed on 19 Feb 2011,available at http : //en.wikipedia.org/wiki/Buddhism − by − country
30. Hinduism by country, accessed on 19 Feb 2011,available at http : //en.wikipedia.org/wiki/Hindusim − by − country
31. Sikhism by country, accessed on 19 Feb 2011,available at http : //en.wikipedia.org/wiki/Sikhism − by − country
32. Judaism by country, accessed on 19 Feb 2011,available at http : //en.wikipedia.org/wiki/Judaism − by − country
33. T. M. Johnson, D. B. Barrett, Quantifying alternate futures of religion andreligions, Futures, 36 (2004) 947-960.34. Brian. J. Grim et. al, Measuring international socio-religious values and conflictby coding U.S. State Department reports, AAPOR, ASA Section on SurveyResearch Methods, (2006) 4120-4127.35. The List:World’s fastest-growing religions, Foreign Policy Magazine, May 14(2007)36. B. Hsu et al., ”Estimating the religious composition of all nations: an empiricalassessment of the World Christian Database, J. Scient. Study of Relig. 47 (4)(2008) 678-693.37. S. Picoli, Jr., R. S. Mendes, Universal features in the growth dynamics of reli-gious activities, Phy. Rev. E. 77 (2008) 036105.38. M. Ausloos, F. Petroni, Statistical dynamics of religions and adherents,ctober 8, 2018 21:22 arxiv12