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Journal of Science and Arts Year 18, No. 1(42), pp. 167-172, 2018
ISSN: 1844 – 9581 Mathematics Section
ORIGINAL PAPER
USING BENFORD’S LAW IN THE ANALYSIS
OF SOCIO-ECONOMIC DATA
DAN-MARIUS COMAN1*, MARIA-GABRIELA HORGA2, ALEXANDRA DANILA2,
MIHAELA-DENISA COMAN3
_________________________________________________
Manuscript received: 12.09.2017; Accepted paper: 22.11.2017;
Published online: 30.03.2018.
Abstract: The article represents a theoretical research which takes into account the
statistical model known as Benford’s Law, followed by a quantitative application-based
research in which we have used Microsoft Excel elements (statistical and graphical functions)
on a set of socio-economic data in order to emphasize how easy it is for a wide range of users
to apply the theoretical concepts mentioned.
Keywords: Benford’s Law, statistical test, spreadsheet
1. INTRODUCTION
The statistical model popularized by Benford [1] in 1938 and developed by Newcomb
[8] is researched and discussed in a wide range of articles and highlights the universality of
applying the method on a set of data such as: electricity bill payment [2], accounting fraud
detection and financial statement audits [7], socio-demographic studies [9]. This research
aims to use the statistical model popularized by Benford in order to show how easy it is to
apply it to a set of data by utilizing a software application (Microsoft Excel) available in the
application package a computer is usually equipped with.
2. OBJECTIVES AND METHODOLOGY OF THE RESEARCH
The article represents a positive research approach, in which the theoretical concept is
reflected and practically demonstrated by an example in the socio-economic field. The
method employed in this research was the consultation of the specialized literature presented
in the bibliography, along with the development and detailing of the theoretical concept by
means of a practical example. Through its practical nature, the analytical procedure based on
the quantitative model of Benford’s law highlights a statistical relationship and falls under the
pragmatic research.
1 Valahia University of Targoviste, Faculty of Economic Sciences, 130004 Targoviste, Romania.
* Corresponding author E-mail: cmnmarius@yahoo.com.
2 Ovidius University of Constanta, Faculty of Economic Sciences, 900470 Constanta, Romania.
E-mail: alexandradanila14@yahoo.com , gabi_horga@gmail.com.
3 Valahia University of Targoviste, Institute of Multidisciplinary Research for Science and Technology, 130004
Targoviste, Romania. E-mail: cmndenisa@yahoo.com.
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3. MATHEMATICAL FOUNDATIONS OF BENFORD’S LAW
Benford’s law refers to the frequency distribution of digits in various situations of
occurrence of numerical data within real-life sources of data. The digit 1 occurs as leading
digit in 30% of the cases, while greater digits occur with smaller probability. The distribution
of first digits is as large as the interval of a logarithmic scale and the results apply to a wide
variety of data. Mathematically, the distribution law applies to base-10 numbers, but there are
applications of the law for the distribution of other numbers in other bases or for the second or
the following digits of a number.
A number string follows Benford’s distribution if the distribution probability of one
digit (di) of the numbers (d=d0d1.....) represented in the in the b-base number system satisfies
the law:
()=1 + 1
,= 1 … ( 1)
(1)
()= log
1 + 1
+,= 0 . . ( 1)
(2)
()= log1 + 1
+ +,
= 0 . . ( 1)
(3)
Numerically, the first digits of a number following Benford’s law have the distribution
in Table 1. Table 1. Data distribution according to Benford’s law.
Digit
Probability
1
0.30103
2
0.17609
3
0.12494
4
0.09691
5
0.07918
6
0.06695
7
0.05799
8
0.05115
9
0.04576
The probability p(d) is proportional to the width of the interval between d and d+1 on
a logarithmic scale, so it will comply with the expected distribution of the mantissa of the
logarithm of that particular number, but not of that number proper, as it is probabilistically
uniformly distributed. The distribution law is named after the physicist Frank Benford who
formulated it intuitively in 1938. According to [3], Benford’s law was explained in several
ways:
• Consequence of the exponential growth process: in principle, it is based on the
assumption that the mantissa of logarithms of numbers is uniformly distributed. This is
correct if the numbers are themselves distributed on multiple orders of magnitude;
• Scalar invariance: the distribution of digits in a real list does not depend on the
unit of measurement used, in other words, multiplication by the same constant will not affect
Using Benford’s Law in … Dan-Marius Coman
ISSN: 1844 – 9581 Mathematics Section
169
distribution. The phenomenon is called scalar invariance and the variables which are log-
normally distributed comply with this property;
• Multiple probability distributions.
The model of Benford’s law applies to datasets which are distributed on several orders
of magnitude. A direct consequence resulting from this observation is that the model is not
valid if we want to check the values in a list (for example bills, payments) between two limit
values (for example between 50,000 and 100,000) or above a minimal value or below a
maximal value.
The particular criteria [10] which limit the application of Benford’s law have been
emphasized on accounting datasets and refer to:
• Assigning numbers for bills, cheques;
• Influencing numbers by human subjective decisions (prices like 9.99);
• Accounts with maximum and minimum limits.
4. PRACTICAL APPLICATION
In addition to applications in the field of fundamental sciences [5] Benford’s law has
found an application in fraud detection in all economic activities. That is why the model has
been included in the CAATS (Computer Auditing Techniques) analytical procedures.
In this article, Benford’s model is applied in order to detect abnormal results in a list
which include data related to the number of people in small towns of the United States of
America [11]. The main goal of applying Benford’s model in the dataset established is
determined by the verification of the following research hypothesis:
“The chosen dataset follows the distribution of Benford’s law.”
The dataset comprises 3,141 observations on the number of people in small towns
located in the United States of America. For space reasons, Fig. 1 presents only a screenshot
of the list used in the analysis.
Figure 1. Screenshot of the dataset.
Using Benford’s Law in … Dan-Marius Coman
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Benford’s law has been applied in Microsoft Excel, which has options (functions,
graphs) to make the necessary calculations. Figures 3 and 4 show the tables in which all the
operations required by this analytical method have been performed.
The calculation of the frequency of occurrence of each digit from 1 to 9 is done in
Table 2.
The formulas which have determined the calculation of the occurrence frequency of
each digit are the following:
• Determining the first digit of numbers in the data series
FLOOR(10^MOD(LOG(Data!A8,10),1),1);
• Creating a contingency table in which every digit determined at the previous point
is represented by the digit 1 on the position corresponding to the digit sequence in the table
header IF(ISERROR($B5) = TRUE,0,IF($B5=3,IF($A5 <= $C$1,1,0),0));
• Determining the sum for each digit in the table header SUM(D5:D4004).
Table 2. Determining the frequency of occurrence of each digit.
Frequency of occurrence of
digits (1 ..9)
992 534 424 260 228 210 173 175 145
Digits 1…9
First digit extracted
from the data series
1 2 3 4 5 6 7 8 9
3
0
0
1
0
0
0
0
0
0
9
0
0
0
0
0
0
0
0
1
2
0
1
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
3
0
0
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
2
0
1
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
3
0
0
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
3
0
0
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
2
0
1
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
4
0
0
0
1
0
0
0
0
0
5
0
0
0
0
1
0
0
0
0
1
1
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
The elements determined at 3 in the previous paragraph help us make the table in
which we shall apply the elements of Benford’s law. The table of analysis is illustrated in
Table 3.
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ISSN: 1844 – 9581 Mathematics Section
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Table 3. Determining the elements of Benford’s law.
Digit Observed
Probability
% of
Observed
Probability
Expected
Probability
(Benford
models)
% of Expected
Probability
(Benford
models)
Variation
(%)
1
992
31.58
946
30.10
4.68
2
534
17.00
553
17.61
-3.58
3
424
13.50
392
12.49
7.45
4
260
8.28
304
9.69
-17.07
5
228
7.26
249
7.92
-9.08
6
210
6.69
210
6.69
-0.13
7
173
5.51
182
5.80
-5.29
8
175
5.57
161
5.12
8.19
9
145
4.62
144
4.58
0.88
Total
3141
100
3141
100
In Table 3, the frequency of occurrence of the first digit has been inserted from Table
2 by copy-paste and, based on this information, we have determined:
• The proportion of elements observed in the total sample, B2/$B$11*100;
• The number of expected elements according to Benford’s law,
LOG((A2+1)/A2,10)*$B$11;
• The proportion of expected elements according to Benford’s law,
(LOG((A2+1)/A2,10))*100;
• The variation of observed elements in relation to the expected elements, ((B2-
D2)/B2)*100.
At a first view of the Table 3, one may note that the greatest variation between the
probability of observed elements and the probability of expected elements according to
Benford’s law manifests for digit 4 (-17.07%). This is also shown by Fig. 2, in which we see
that the statistical distribution of the first digit follows the theoretical distribution in a similar
manner; we should, however, mention that in the case of the digit 4 there is a greater variation
between statistical distribution and theoretical distribution.
Figure 2. First digit vs. Benford’s Law.
31.58
17.00
13.50
8.28
7.26
6.69
5.51
5.57
4.62
30.10
17.61
12.49
9.69
7.92
6.69
5.80
5.12
4.58
-3.00
2.00
7.00
12.00
17.00
22.00
27.00
32.00
1
2
3
4
5
6
7
8
9
Firstdigit in county population vs. Benford's Law
% of Observed Probability
% of Expected Probability (Benford model)
Using Benford’s Law in … Dan-Marius Coman
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172
The observation makes us consider the research question: the chosen dataset follows
the distribution of Benford’s law, which we can transpose in the following hypothesis:
(1) H0: The first digit of the dataset on the observed population follows the theoretical
distribution of Benford’s law
(2) H1: The first digit of the dataset on the observed population does not follow the
theoretical distribution of Benford’s law
To test this hypothesis, we shall use the chi-square test [4] which allows the
comparison between two distributions (observed vs. theoretical) with a view to establishing
either an independence between them or their homogeneity. The chi-square test formula is:
= ( )
(4)
The application of the calculation formula was done in table 4, obtaining a value of
15.43. The obtained value is interpreted by comparing it with the corresponding value in the
chi-square table. Since in this case there are nine variants, the number of degrees of freedom
is c-1, i.e. 8.
Table 4. Determining the elements of the chi-square test.
Digit Observed
Probability
% of
Observed
Probability
Expected
Probability
(Benford
models)
% of Expected
Probability
(Benford
models)
Variation
(%) Chi
square
1
992
31.58
946
30.10
4.68
2.28
2
534
17.00
553
17.61
-3.58
0.66
3
424
13.50
392
12.49
7.45
2.54
4
260
8.28
304
9.69
-17.07
6.47
5
228
7.26
249
7.92
-9.08
1.72
6
210
6.69
210
6.69
-0.13
0.00
7
173
5.51
182
5.80
-5.29
0.46
8
175
5.57
161
5.12
8.19
1.28
9
145
4.62
144
4.58
0.88
0.01
Total
3141
100
3141
100
15.43
The chi-square value of 15.43 is less than that in the Table 4 (15.5073) at 8 degrees of
freedom and a significance threshold p = 0.9847, calculated based on the Microsoft Excel
function CHISQ.DIST(G11,8,TRUE).
Based on chi-square distribution, the probability to reject the null hypothesis is
calculated. Usually the research question is accepted if the probability (p) of rejecting the
hypothesis H0 is less than 5%. In the context of p=0.9847 > 0.05, which denotes that there are
no significant differences between the observed distribution and the theoretical one and
there is no sufficient evidence to believe that the variation observed in the case of digit 4
between the two distributions may be due to the occurrence of a mystification (alteration) of
data reporting regarding the town population.
In the specialized literature [6] a number of problems are reported when comparing
observed and theoretical distributions. Among the observations noted are: issues regarding the
choice of the number of classes, issues on the width of frequency classes, issues regarding the
number of observations inside each frequency class. All these elements lead, in the case of
Benford’s law, to the calculation of an additional test called mean absolute deviation (MAD),
which measures the average of absolute deviation of the frequencies of each digit from
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ISSN: 1844 – 9581 Mathematics Section
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Benford’s ideal frequency. In this application, MAD is calculated in Table 5 based on the
following formula:
= | |
(5)
The value obtained by applying the formula may fall into three classes according to
the following algorithm: 0.000 < MAD >0.006 – close conformity; 0.006 < MAD < 0.012 –
acceptable conformity; 0.012 < MAD < 0.015 – marginal conformity; MAD > 0.015 – non
conformity.
Table 5. MAD Calculation
Digit Observed
Probability
% of
Observed
Probability
Expected
Probability
(Benford
models)
% of
Expected
Probability
(Benford
models)
Variation
(%) Chi
square MAD
1
992
31.58
946
30.10
4.68
2.28
1.48
2
534
17.00
553
17.61
-3.58
0.66
0.61
3
424
13.50
392
12.49
7.45
2.54
1.01
4
260
8.28
304
9.69
-17.07
6.47
1.41
5
228
7.26
249
7.92
-9.08
1.72
0.66
6
210
6.69
210
6.69
-0.13
0.00
0.01
7
173
5.51
182
5.80
-5.29
0.46
0.29
8
175
5.57
161
5.12
8.19
1.28
0.46
9
145
4.62
144
4.58
0.88
0.01
0.04
Total
3141
100
3141
100
15.43
0.6625
Following MAD calculation, the value 0.6625 is classified as non-conformity, which
supports the idea that there are no significant differences between observed and theoretical
distributions, which has also been shown by the chi-square test.
5. CONCLUSIONS
Benford’s law is a powerful tool, integrated into dedicated applications (e.g. Idea
Casewear), for auditors in identifying the risk of economic fraud, but the example presented is
not intended to highlight abnormal aspects in reporting socio-economic data on the number of
inhabitants of towns in the USA. This paper aims to present, in a simple way, how to use an
easy working tool in order to calculate specific elements of Benford’s law: chi-square test,
mean absolute deviation, the graph of correlation of the researched data series with the
expected data series.
For regular users, beginning practitioners, who do not have access to specialized
applications, the example presented may be a good start in acquiring the skills of using
Microsoft Excel spreadsheet to apply statistical tests for fraud detection in judicial accounting
expertise or in auditing.
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REFERENCES
[1] Benford, F., Proceedings of the American Philosophical Society, 4(78), 551, 1938.
[2] Christian, C., Gupta, S., Lin, S. M., National Taxa Journal, 4(46), 487, 1993.
[3] Coracioni, A., Audit Financiar, 11(104), 23, 2013.
[4] Florea, N.V., Mihai, D.C., Journal of Science and Arts, 1(38), 81, 2017.
[5] Jäntschi, L., Bolboaca, S., Stoenoiu, C., Bulletin UASM Agriculture, 66(1), 82, 2009.
[6] Jäntschi L., Proceedings of International Conference on Recent Achievements in
Mechatronics, Automation, Computer Science and Robotics, 239, 2011.
[7] Marcini, S., Hamilton T., Journal of the Risk and Uncertainty, 1(32), 57, 2006.
[8] Newcomb, S., American Journal of Mathematics, 4, 39, 1881.
[9] Sandron, F., Population, 57(4-5), 755, 2002.
[10] Shalini, T., Kinjal, M., IOSR Journal of Economics and Finance, 1-9, 2014.
[11] https://introductorystats.wordpress.com/2011/11/25/benfords-law-and-us-census-data-
part-ii/," last accessed 27.01.2018.
