Modelling Conformity of Nigeria s Recent Population Censuses With Benford s Distribution

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1 International Journal Of Mathematics And Statistics Invention (IJMSI) E-ISSN: P-ISSN: Volume 3 Issue 2 February PP Modelling Conformity of Nigeria s Recent Population Censuses With Benford s Distribution N. A. IKOBA 1 and E. T. JOLAYEMI 2 1,2, Department of Statistics, University of Ilorin, Ilorin, Nigeria ABSTRACT : Benford's distribution, a probability distribution which was discovered at the twilight of the 19 th century, can be used as a robust tool in exposing error and/or fraud in random data in various scenarios. Because of the massive political leveraging involved in population census results in Nigeria, the census exercise has been open to manipulation and distortion. In this paper, we analyze the distribution of the first significant digits of the 1991 and 2006 Nigerian population censuses to establish conformity with Benford's distribution. We also analyze the aggregate census data for the six geo-political zones of the country to determine the level of dispersion of the distribution of first digits of the census counts. Our analyses showed that the North-West region had the highest dispersion for both the 1991 and 2006 census (and by extension, the highest level of nonconformity with Benford s law) while the North-East and South-West had the lowest dispersion for the 1991 and 2006 censuses, respectively. KEY WORDS: Benford s distribution; Chi-Square test; First Significant Digits (FSD); scale-invariance. I. INTRODUCTION The history of the conduct of population censuses in Nigeria is replete with accusations and counteraccusations from various sections of the country that the results of such censuses are flawed. Because the results from any census have far-reaching political and socio-economic implications, it has become possible for government officials to be influenced through pecuniary and other considerations, to tamper with and alter authentic counts by inflating these figures. This has brought a huge credibility problem for the census organizing body in Nigeria, the National Population Commission (NPC), which even led to the resignation of the head of the organization, when he alluded that the last national population census figures in the country was not sufficiently credible. While various in-house measures have been put in place to curtail the excesses of census officials in the execution of their duties during the forthcoming national census of 2016, there is the need to evolve sufficiently robust measures to test the validity of census figures, using global trends and practices. We seek to test the validity of the last two national population censuses held in Nigeria. For this purpose, we use the Benford s distribution of first significant digits to test conformity of the 1991 and 2006 population census figures with what should be expected if the distribution of first significant digits of both censuses follow Benford's distribution. We shall use the Chi-square test to measure the goodness-of-fit of the data to Benford's distribution. We also seek to examine the level of conformity to Benford's distribution of the aggregate data, consisting of the six geo-political regions of the country in order to establish which region contributed more to the dispersion, if any. In section 2, we present a formal description of Benford's Distribution and provide some of its properties, as well as a review of relevant literature. Our results are presented and discussed in section 3. Finally, our conclusions and recommendations are expressed in section 4. II. BENFORD S DISTRIBUTION AND ITS JUSTIFICATION Benford's distribution of first significant digits is given by Distributions of successive digits and joint distributions can also be specified. For any specific sequence of leading digits n, the probability of occurrence is given by equation (2.1). Thus, the probability distribution for successive digits is the sum of the probabilities associated with all possible combinations of that number length ending in digits 0 through 9. 1 P a g e

2 As we move farther from the first significant digits, the digits tend towards being equally distributed. Dealing with numbers with three or more digits, for all practical purposes, the ending digits are expected to be uniformly distributed [1]. The first significant digit phenomenon was first observed by the astronomer and mathematician, Simon Newcomb in 1881 [2]. Having observed the pages of books of logarithms at the library (the early pages were quite dirty than the others, showing greater usage of the early pages), Newcomb came to the conclusion that the probability that a number has a particular first significant digit (that is, first non-zero digit) d can be calculated as follows: Newcomb's article was corroborated 57 years later by Frank Benford [3], a physicist working with General Electric. Benford, seemingly working independently, made the same observations about logarithm tables and came to the same conclusion as Newcomb. He then tested his conjecture with an effort to collect data from as many fields as possible and to include a wide variety of types. An analysis of the numbers from different sources showed that the numbers taken from unrelated subjects, such as group of newspaper items, show a much better agreement with the log distribution, than do numbers from mathematical tabulations or other formal data. He claimed that the Law of Anomalous Numbers, as he then called the law, is a general probability law of widespread application. It was shown that the series of 1-day returns on the Dow Jones Industrial Average index and the Standard and Poor's index reasonably agree with Benford's law, and therefore belong to the family of anomalous or outlaw numbers [4]. The analysis presented suggested that small changes are more likely than large ones; at the same time the closer the daily changes are (in absolute value) to 0.1%, the more probable they are too. The authors in [5] developed statistical tests that can be used by auditors as analytical procedures in the planning stages of the audit. A general description of the mathematical foundations of the first digit phenomenon was undertaken in [2], and some recent applications were reviewed. A major mathematical property of Benford's distribution is that it is the only probability distribution that is scale-invariant and the only one that is base-invariant (excluding the constant 1). The author also established that a surprisingly diverse collection of empirical data obey the law, and that since Benford's popularization of the law, an abundance of empirical evidence have appeared and these include tables of physical constants, numbers appearing on newspaper front pages, accounting data, scientific calculations, stock market closing figures, etc. It was shown in [6] that the populations of 3,141 counties in the 1990 United States census showed a good fit for Benford's law. The actual proportions follow Benford's law quite closely, which is what would be expected from authentic, unmanipulated data. Mark Nigrini has amassed extensive empirical evidence of the occurrence of Benford's law in many areas of accounting and demographic data and came to the conclusion that in a wide variety of accounting situations, the significant-digit frequencies of the true data conformed very closely to Benford's law. When people fabricate data, on the other hand, either for fraudulent purposes or just to fill in the blanks, the concocted data rarely conforms to Benford's law. That people cannot act truly randomly, even in situations where it is to their advantage to do so, is a well-established fact of psychology [2]. The law has been proposed for use by the U.S. Internal Revenue Service and in detecting accounting fraud [2]. The authors in [7] quantified compliance with Benford's law for several survival distributions, with emphasis on probability distributions that obey Benford's law. They conjectured that the reason that Benford's law applied to so many datasets may simply be due to the fact that many popular parametric lifetime models closely follow the law for particular values of their parameters. Benford's law was applied to identify inaccurate survey data in [8]. The method involved examining the distribution of the leading digits of all the numbers reported on a survey form. By knowing the distribution of the leading digits, one can identify unusual data which may be fraudulent or generated by an error-prone process.it was established in [9], that Benford s distribution and the uniform distribution belong to a family arising from mixtures of uniforms, and characterized the first significant digit patterns for a one-parameter subset of the family. The family members exhibit decreasing first significant digit probabilities. The empirical analysis suggested that although the uniform first significant digit pattern and Benford's law are reasonable models for some data, alternative family members better fit other data. His objective was to provide an analytical framework that allows comparison of an observed first significant digit pattern against a wide variety of one-parameter theoretical first significant digit distributions that belong to the larger family of mixtures of uniform distributions.the authors in [10] explored whether crime statistics are Benford-distributed. Using historical data from several security agencies in the United States of America, the authors utilized first, second, first-two, first-three, and last-two digits analyses to assess the level of conformity of observed data to Benford's expected distributions, and obtained close conformity. 2 P a g e

3 Upon disaggregation of the data, it was discovered that violent offences (particularly murder, manslaughter, rape and robbery) generally show less conformity to Benford s than do property offences (burglary, larceny, excepting motor vehicle thefts).benford s law was applied in [11] to carry a statistical analysis of the 2009 Albanian parliamentary elections, exposing areas of non-conformity with the law. The conclusion was that fraud had been perpetuated in those areas, based on non-conformity with Benford s law. Some recent applications of the significant-digit law include testing of mathematical models; computer applications to minimize storage space and maximize rate of output; detection of fraud in financial documents; etc. Not all datasets follow Benford's law. Those datasets most likely to conform to Benford's distribution will have the following characteristics [6]: [1]. The numbers describe the sizes of similar phenomena (for example, market value of corporations). [2]. The numbers do not contain a built-in maximum or minimum value. The scale-invariance property of Benford's law helps us to understand why the law works on financial data throughout the world, even though the data are expressed in different currencies. Hill [2] has formulated a new statistical form of the significant-digit law. The theorem which is called Random samples from random experiments theorem, states that: If distributions are selected at random (in any unbiased way) and random samples are taken from each of these distributions, then the significant-digit frequencies of the combined sample will converge to Benford's distribution, even though the individual distributions selected may not follow the law. III. RESULTS AND DISCUSSION The data used for this study were extracted from the Annual Abstract of Statistics, 2009, published by Nigeria s National Bureau of Statistics (NBS) [12]. Census counts of the 592 Local Government areas of Nigeria were used to compute the distribution of the first significant digits for the 1991 census. For the 2006 census, results from the 774 local government areas of the country were used to compute the distribution of the first significant digits. The goodness-of-fit test deployed to test conformity with Benford's Law was the chisquare test. The chi-square test compares the actual counts of the census data with the expected counts, which follows the hypothesized distribution (Benford's). The null hypothesis is that the first digits of the data follow Benford's distribution. The chi-square statistic for the test is given by where and represent the observed count and the expected count, respectively of the i th digit. The decision rule for the test is to reject the null hypothesis at the 5% level of significance ( ) if if the p-value is less than the level of significance,. The p-value (or probability value) is the probability of getting a sample statistic in the direction of the alternative hypothesis when the null hypothesis is true. In other words, the p-value is the actual area under the curve of the distribution representing the probability of a particular sample statistic occurring if the null hypothesis is true [13]. The Kolmogorov-Smirnov (K-S) test can equally be applied. This was done and identical results obtained. The aggregate data for the six geo-political zones of Nigeria, consisting of the South-East (SE), South-South (SS), South-West (SW), North-East (NE), North-West (NW), and North-Central (NC) were also analyzed. It should be noted that for ease of analysis, the Federal Capital Territory (FCT) was lumped into the North-Central zone. The disaggregation enabled us to analyze conformity with Benford's Law across the geo-political zones of Nigeria. Figure 3.1 shows the actual counts of first significant digits (FSD) compared to the expected counts for the 1991 census data, assuming Benford's distribution. Figure 3.2 shows the corresponding counts for the 2006 census data. Figure 3.3 shows the cumulative density function of the first significant digits for the 1991 and 2006 census data, as well as that of Benford's distribution. Table 3.1 shows the distribution of first significant digits for the 1991 and 2006 census data, as well as for Benford's distribution. The table also shows the corresponding cumulative proportions. Table 3.2 shows the distribution of first significant digits for the six geopolitical regions for the 1991 census, with table 3.3 showing the corresponding data for the 2006 census. Table 3.4 gives the results from the chi-square test for the 1991 and 2006 census data, showing the overall and the aggregate results for the geo-political zones. To be clear, Benford's law cannot deduce intention, it can only be used to detect unusual or unexpected data. These unusual or unexpected data may or may not have been an intentional product, but the technique of digital analysis is blind to the underlying intention and simply highlights possible irregularities [10]. Digital analysis could help support investigative efforts, but it is not a substitute for a thorough investigation. Since human choices are not random, invented numbers are unlikely to follow Benford's law [6]. or 3 P a g e

4 Table 3.1: Comparison of the distribution of first significant digits for 1991 and 2006 Censuses with Benford's Law FSD 1991 Census 2006 Census Benford s Law Key: FSD First Significant Digit; C f Cumulative frequency Table 3.2: Proportion of First Significant Digits for the six geo-political zones, 1991 Census FSD SE SS SW NE NW NC Benford s Law Table 3.3: Proportion of First Significant Digits for the six geo-political zones, 2006 Census FSD SE SS SW NE NW NC Benford s Law Figure 3.1: Observed and expected first digits counts, 1991 Census. 4 P a g e

5 Figure 3.2: Observed and expected first digits counts, 2006 Census. Figure 3.3: Plot of the Cumulative Density function of Benford's Law and 1991, 2006 census data. Table 3.4: Result of the Chi-Square tests for 1991 and 2006 Census data A preliminary study of the data shows lack of conformity with Benford's Law, as could be seen from figures 3.1 and 3.2, with the digit 1 accounting for more than 50% for both the 1991 and 2006 census data. The analyses of the distribution of first significant digits for the 1991 census data show that the null hypothesis that the overall data follows Benford's distribution is rejected at the 5% level of significance, using the chi- 5 P a g e

6 square test. The disaggregated data for the six geo-political zones of the country also show non-conformity with Benford's distribution at the 5% level of significance. A careful look at the results of the chi-square test for the 1991 census data reveals that the North-East region produced a p-value of It would seem that the level of malpractice in the North-East during the 1991 census had less impact on the result for that zone, in comparison to the other geo-political regions. For the 2006 census data, the chi-square tests showed that both the overall data and the aggregate data for the six geo-political zones were significant at the 5% significance level. Hence the data exhibited significant non-conformity to Benford's distribution. Looking at the aggregate data for 1991 census, it can be seen that the North-East geo-political zone had the least value (16.976), accounting for 9% of the overall dispersion, while the North-West region had the largest value (84.236), which is about 45% of the overall dispersion. However, the Southern part of the country (consisting of South-East, South-South and South-West) accounted for 56.4% of the overall dispersion compared to 43.6% for the Northern part (North-East, North-West and North-Central). The aggregate data for the 2006 census revealed that the South-West geo-political zone had the lowest value (37.076), which is 9.6% of the overall dispersion, while the North-West region again had the highest value ( ), accounting for 30% of the overall dispersion. The Southern region accounted for 47.7% of the overall dispersion, while the Northern region accounted for 52.3%.When the result of the chi-square tests for the 1991 census data is compared with that of the 2006 census data (table 3.4), it would seem that there had been a massive shift away from the expected distribution, as the computed value doubled. Obviously, it may be that a more extensive level of manipulation of the census figures was carried out during the 2006 census exercise. There have been stories flying around of how census officials were induced financially by community leaders and politicians to inflate the counts of the concerned community. These anomalies were not localized in only a single region, but happened across the country. It might also not be unconnected with the fact that the number of Local Government Areas had increased from 592 in 1991 to 774 in This was as a result of the creation of additional six states (Bayelsa, Ebonyi, Ekiti, Gombe, Nasarawa and Zamfara) from the existing states of the Nigerian federation. If indeed there was manipulation, the dispersion from the expected distribution will increase with the addition of more states and local governments. The first significant digit 1 accounted for 56% of the counts for the 1991 census data. If it were to follow Benford's distribution, it would have been about 30%. Upon closer scrutiny of the distribution of first significant digits for the 1991 census data, we see that digits 6, 7, 8 and 9 had proportions that were very close to that of Benford's distribution but digit 1 exhibited a marked departure from Benford's, which in turn, impacted on the proportions for digits 2, 3, 4 and 5. Thus the degree of imputation was highly skewed in favour of the first significant digit 1, to the detriment of the other leading digits. Similarly for the 2006 census data, digit 1 accounted for 56% of the overall counts, while digits 8 and 9 had proportions very close to the corresponding Benford's proportion. With this data, the distribution of the first significant digits was heavily skewed towards 1 and 2, as the two digits accounted for almost 80% of the overall counts. This might give us an inkling of the nature of manipulation. It is obvious that the manipulation of the true census count (if indeed there was manipulation) was mainly in 1's and 2's. This means that, for example, an original census count of 85,000 was changed to a value whose first digit is most likely to be 1 or 2. A look at the aggregate data for both censuses (tables 3.2 and 3.3) also revealed that the first two digits 1 and 2 accounted for more than 50% of the counts, except interestingly for the North-East in the 1991 census, which has a cumulative proportion of which is exceptionally close to what is expected for Benford's distribution (0.477). When the geo-political zones were ranked according their value, the North-West region had the highest value for both the 1991 and 2006 censuses, while the North-East and South-West had the lowest for the 1991 and 2006 census, respectively. IV. CONCLUSION AND RECOMMENDATION From the evidence of the data used for this study, it appears that fraudulent and random-guess data tend to have far too many numbers beginning with 1 and far too few numbers beginning with 6.There was significant non-conformity to Benford's distribution for the two censuses as well as the aggregate data, except for the North-east geo-political region, whose 1991 analysis showed that the distribution of first significant digits follow Benford's distribution. Fitting Benford's distribution on population census data may serve as one of the quality control tools to assert the level of authenticity of census results anywhere. A more comprehensive analysis may be carried out to test conformity with Benford's distribution of the first two or three significant digits as a means of further ascertaining the validity of census data. 6 P a g e

7 This is therefore a worthy quality assessment tool that should be looked into by the Population census commissions in census preparations. REFERENCES [1.] Nigrini, M. J. (2011). Forensic Analytics. John Wiley & Sons Inc. [2.] Hill, T. P. (1998). The First Digit Phenomenon. American Scientist, vol. 86, No. 4, pp [3.] Benford, F. (1938). The Law of Anomalous Numbers. Proceedings of the American Philosophical Society, vol. 78, No. 4, pp [4.] Ley, E. (1996). On The Peculiar Distribution of The U.S. Stock Indexes Digits. The American Statistician, vol. 50, No. 4, pp [5.] Nigrini, M. J. and Mittermaier, L. J. (1997). The Use of Benford's Law as an Aid in Analytical Procedures. Auditing: A Journal of Practice and Theory, vol. 16, No. 2, pp [6.] Nigrini, M. J. (1999). I've Got Your Number. Journal of Accountancy, 187 (5), pp [7.] Leemis, L. M.; Shmeiser, B. W.; and Evans, D. L. (2000). Survival Distributions Satisfying Benford's Law. The American Statistician, vol. 54, No. 4, pp [8.] Swanson, D.; Cho, M. J. and Eltinge, J. (2003). Detecting Fraudulent or Error-Prone Survey Data Using Benford's Law. Proceedings from 2003 Joint Statistical Meetings - Section on Survey Research Method, pp [9.] Rodriguez, R. J. (2004). First Significant Digit Patterns From Mixtures of Uniform Distributions. The American Statistician, vol. 58, No. 1, pp [10.] Hickman, M. J. and Rice, S. K. (2010). Digital Analysis of Crime Statistics: Does Crime Conform to Benford's Law? Journal of Quantitative Criminology, vol. 26, No. 3, pp [11.] Berdufi, D. (2013). Statistical Detection of Vote Count Fraud: 2009 Albanian Parliamentary Election and Benford's Law. Academic Journal of Interdisciplinary Studies, vol. 2, No. 8, pp [12.] Annual Abstract of Statistics, Published by the National Bureau of Statistics, [13.] Bluman, A. G. (2012). Elementary Statistics: A Step by Step Approach. McGraw-Hill, New York. 7 P a g e