The power of one: Benford's Law

Abstract

The concept of Benford's law, also known as the first-digit phenomenon, has been known to mathematicians since 1881. It is counter-intuitive, difficult to explain in simple terms, and has suffered from being described variously as 'a numerical aberration', 'an oddity', 'a mystery' - but also as 'a mathematical gem'. However, it has developed into a recognised statistical technique with several practical applications, of which the most notable is as a fraud detection mechanism in forensic accounting. This paper will briefly discuss and demonstrate the special numerical characteristics of Benford's law. It will attempt to investigate the law's possible application to the detection of data manipulation and data tampering that might exist in papers published in engineering and scientific journals. Firstly, it will be applied to an investigation of the so-called Fisher-Mendel controversy. Secondly, Benford's analysis will be applied to six recently published papers selected from the South African Journal of Industrial Engineering.

Full text

South African Journal of Industrial Engineering August 2017 Vol 28(2), pp 1-13
1
THE POWER OF ONE: BENFORD’S LAW
P.S. Kruger1* & V.S.S. Yadavalli1
ARTICLE INFO
Article details
Submitted by authors 28 Mar 2017
Accepted for publication 4 Jul 2017
Available online 31 Aug 2017
Contact details
* Corresponding author
paul.kruger12@outlook.com
Author affiliations
1 Department of Industrial and
Systems Engineering, University of
Pretoria, South Africa
DOI
http://dx.doi.org/10.7166/28-2-1753
ABSTRACT
The concept of Benford’s law, also known as the first-digit
phenomenon, has been known to mathematicians since 1881. It is
counter-intuitive, difficult to explain in simple terms, and has
suffered from being described variously as ‘a numerical aberration’,
‘an oddity’, ‘a mystery’ – but also as ‘a mathematical gem’.
However, it has developed into a recognised statistical technique
with several practical applications, of which the most notable is as
a fraud detection mechanism in forensic accounting. This paper will
briefly discuss and demonstrate the special numerical
characteristics of Benford’s law. It will attempt to investigate the
law’s possible application to the detection of data manipulation and
data tampering that might exist in papers published in engineering
and scientific journals. Firstly, it will be applied to an investigation
of the so-called Fisher-Mendel controversy. Secondly, Benford’s
analysis will be applied to six recently published papers selected
from the South African Journal of Industrial Engineering.
OPSOMMING
Die konsep van Benford se wet, ook bekend as die eerste-syfer-
fenomeen, is bekend aan wiskundiges sedert 1881. Dit is teen-
intuïtief, moeilik om te verduidelik op ’n eenvoudige wyse, en gaan
gebuk onder verskeie beskrywings soos ‘’n numeriese afwyking’, ‘’n
koddigheid’, ‘’n misterie’, maar ook as ‘’n wiskundige juweel’. Dit
het nietemin ontwikkel in ’n erkende statistiese tegniek met vele
praktiese toepassings, waarvan die gebruik as ’n bedrog
betrappingsmeganisme in forensiese rekeningkunde noemens-
waardig is. Hierdie artikel sal die spesiale numeriese karakteristieke
van Benford se wet bespreek en demonstreer. Dit sal die wet se
moontlike gebruik om datamanipulering en -vervalsing wat mag
bestaan in ingenieurs- en wetenskaplike publikasies te identifiseer.
Eerstens sal dit toegepas word om die sogenaamde Fisher-Mendel
kontroversie te ondersoek. Tweedens sal dit gebruik word om ses
artikels wat onlangs gepubliseer is in die Suid-Afrikaanse Tydskrif
vir Bedryfsingenieurwese aan ’n Benford analise te onderwerp.
First with the head, then with the heart, you'll be ahead from the start.
From The power of one by Bryce Courtenay
1 INTRODUCTION
We are drowning in information but starved for knowledge. John Naisbitt
Numbers are an inescapable part of everyday life, and terms such as data processing, data capture,
database, data mart, data warehouse, data mining, data farming, metadata, and even big data,
have become almost household words. Numbers are used for many purposes such as counting,
measuring, reporting, accounting, mathematics, labelling, ordering, and coding. Technology,
especially computer technology, has caused an explosion in the amount of readily available – but

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sometimes unorganised – data. The challenge is to change the almost overwhelming amount of
available data and numbers into information and insight. This is in many ways the main purpose of
descriptive statistics – that is, to provide tools to analyse, model, and identify the possible existence
of usable patterns in a data set. However, the identification and isolation of such patterns can often
be difficult without special computational tools or extraordinary perception. This can be
demonstrated by what occurred during a meeting in 1919 between the two great numerical
mathematicians Srinivasa Ramanujan and G.H. Hardy [1]. The following anecdote has been
recounted by Hardy on several occasions: Once, in a taxi from London on his way to visit Ramanujan
in hospital, Hardy noticed the taxi’s number, 1729. He must have thought about it a little because
he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his
disappointment with the number. It was, he declared, “rather a dull number”, adding that he hoped
that it was not a bad omen. “No, Hardy,” said Ramanujan, “it is a very interesting number. It is the
smallest number expressible as the sum of two [positive] cubes in two different ways” [1]. These
numbers became known as ‘taxicab-numbers’, and since 1919 only six such numbers have been
identified. The largest and most recent was discovered in 2008, and contains 23 digits.
Unfortunately, most humans do not possess the extraordinary mathematical vision and insight of
Ramanujan, and so must rely on the proper application of the available statistical techniques. One
such technique is known as Benford’s law, or the first-digit phenomenon. This paper will attempt to
discuss and illustrate the characteristics and application of Benford’s law.
2 BENFORD’S LAW
In fact, 'the law is an ass'. From Revenge for honour by George Chapman
Benford’s law is well-known among mathematicians, statisticians, and accountants, and recently
several articles have appeared, including in the popular media [1 - 9]. However, it is often perceived
as no more than an interesting mathematical oddity. Given the number and variety of data sets that
might conform to Benford’s law, it is somewhat surprising that there are not many more applications,
apart from forensic accounting. Some possible applications have been mentioned or suggested, such
as analysing election results, digital signal processing, digital analysis of data integrity, information
technology auditing, accounts receivable, credit card transactions, loan data, stock prices, purchase
orders, and inventory [1, 5, 9, 10]. However, Benford’s law remains an enigma, and continues to
defy attempts at an easy derivation [10].
2.1 The basic principles of Benford’s law
There are three types of lies: lies, damned lies, and statistics. Benjamin Disraeli
Consider the generation of 1000 random numbers between 1 and 1000, using a reliable pseudo-
random number generator. If the first significant digit of each random number is isolated and
classified as one of the numbers 1 to 9 and counted, common sense and intuition would indicate
that each number between 1 and 9 should appear with approximately the same probability or
relative frequency. This is true, as indicated by Figure 1. However, if 10 such random number
streams are generated, multiplied by each other, and subjected to the same numerical
manipulation, the relative frequency histogram shown in Figure 2 is the result. This is certainly not
the rectangular distribution shown in Figure 1, and is known as Benford’s law, or the first-digit
phenomenon [1-9]. Furthermore, if the 10 random numbers are added rather than multiplied, the
relative frequency histogram shown in Figure 3 is the result. This distribution seems to be close to
a normal distribution, and is probably caused by the central limit theorem – which might be an
indication that Benford’s law is similar to that theorem [11]. The number 10 was chosen after
preliminary experiments had shown that this is adequate to show the emerging patterns clearly.
Simply stated, Benford’s law claims that for many, but not all, data sets with a natural origin,
including the results from mathematical operations, might produce relative frequencies for the first
digit where the occurrence of the smaller numbers is higher than that of the larger numbers [8].

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Figure 1: Histogram of the
first digit of a stream of
single random numbers
Figure 2: Histogram of the
first digit of the product of
10 random number streams
Figure 3: Histogram of the
first digit of the sum of 10
random number streams
2.2 The background to Benford’s law
Those who do not remember the past are condemned to repeat it. George Santayana
In 1881 the astronomer Simon Newcomb noticed that some of the pages in his book of logarithmic
tables were much more worn and dirty than the other pages. Furthermore, the numbers appearing
in these pages tended to start with ‘1’. He published a paper to report on his observations [12], but
this paper was largely ignored and forgotten. In 1938 the physicist Frank Benford rediscovered this
phenomenon, and published a paper that referred to it as the “law of anomalous numbers” [2]. In
this paper, Benford investigated 20 datasets from a variety of sources and origins – for example, the
surface area of rivers, the size of the population in cities, numbers appearing in the Reader’s Digest,
and the results obtained from mathematical operations such as power functions [2]. Neither
Newcomb nor Benford explained the phenomenon, but both suggested that the resultant distribution
might be of a logarithmic type. It was only in 1995 that a statistical derivation of Benford’s law was
published [11], showing that the distribution of the first digit was indeed a logarithmic series
distribution given by the following expression:
P(D = d) = Log10(1+1/d)
The variable D is the first digit having values equal to d = 1, 2 …. 9
The histogram of this distribution is shown in Figure 4. Throughout the rest of this paper, this
distribution provides the frequencies that are expected when Benford’s law is considered applicable.
Figure 5 shows the comparison between the histograms of the product of 10 random numbers and
Benford’s law. Tables 1 and 2 show the results of statistical tests performed for both the product
and the sum of 10 random digits. These tests will be described in section 2.3. It seems as if
multiplication operations provide a good conformance with Benford’s law, but not addition. One of
the main applications of Benford’s law is in forensic accounting [3, 4, 6], where it is used to detect
possible fraud. However, financial statements often contain a significant number of addition
operations. This apparent contradiction is typical of Benford’s law, since it often displays exceptions
that are difficult to explain.
P(D = d) = log10(1+1/d)
Figure 5: Comparison between the
histograms of Benford’s law and the
product
Figure 4: Histogram of
Benford’s law

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Table 1: Statistical analysis of the results for the product of 10 random numbers
Number
1
2
3
4
5
6
7
8
9
Observed frequency
of the first digit
314
152
119
89
79
54
75
70
48
Chi-square P-value = 0.16
Cannot reject H0
Sample size = 1000
RMSE-fit index = 0.014
Fit is very good
Tests for proportions
P-value =
0.56
0.56
0.84
0.99
0.45
0.47
0.34
0.55
0.65
Reject H0?
No
No
No
No
No
No
No
No
No
2.3 Evaluating the conformance to Benford’s law
Get your data first, then you can distort them at your leisure.
Attributed to Mark Twain (Samuel Langhorne Clemens)
The graphical evidence of conformance provided by Figure 5 might be significant, compelling, and
even dominating. However, several statistical tests known as the goodness-of-fit or lack-of-fit tests
[14], and techniques based on so-called fit indexes [14-17], are available for supporting the graphical
evidence. Apart from the graphical evidence, only three such techniques will be used in this paper.
The chi-square goodness-of-fit test is one of the best-known and most widely-used goodness-of-fit
tests, although it does suffer from some limitations [13]. It is sensitive to sample size and outliers,
and does not provide much evidence for the strength of the fit – although the magnitude of the P-
value might be useful. The chi-square tests will be conducted using the following hypothesis
statement:
Null hypothesis H0: The fit between the observed and expected frequencies is good
Alternative hypothesis HA: The fit is poor
The root mean square error (RMSE) fit index [14-17] is considered as one of the best indexes of its
kind [16] and is easier to understand and evaluate than the chi-square test. Furthermore, it may be
used to evaluate the strength of the fit and is useful for comparing different data sets.
Both the chi-square test and the RMSE-fit index evaluate the fit between the observed and expected
frequencies in its entirety. A hypotheses test for the difference in proportions may be used to
evaluate the difference in each pair of relative frequencies separately [13]. This may proof valuable
in determining which pair of relative frequencies contributes the most to the possible failure of the
chi-square test and/or the RMSE-fit index. Furthermore. It may provide a suitable starting point for
any further investigation that may be considered. The tests for the difference in proportions will be
conducted using the following hypothesis statement:
Null hypothesis H0: p1 = p2 The fit between the observed and expected frequencies is good
Alternative hypothesis HA: p1 ≠ p2 The fit is poor
where p1 and p2 are the observed and expected relative frequencies respectively.
For most statistical hypothesis tests, it is necessary to define a level of significance, the value of
which might be open to debate. The great Swiss mathematician, Jacob Bernoulli, who could be
considered the initiator of the concept of statistical inference, referred to the level of significance
as “the level of moral certainty” [18]. Bernoulli professed to be unsure what an acceptable value
for the level of moral certainty should be and, given his background in the law, suggested: “It would
be useful if the magistrates set up fixed limits for moral certainty” [18]. He derives his definition of
probability from previous work by Gottfried Wilhelm Leibniz, and concluded: “Probability is a degree
of certainty” [18] – and that is what a level of significance is. This supports the notion that an
appropriate value for a probability might be subject to the situation, personal judgment, and risk
preference. A value of the level of confidence of 0.05 was chosen for the purposes of this paper, as
it is the most widely-used and widely-accepted value, and will be used to evaluate the P-values (as
shown in Table 3). However, rejecting a hypothesis based on a level of significance of 0.05 might be
unnecessarily conservative in the case of Benford’s law. Based on several references from the
literature [14-17], the cut-off criterion for the value of the RSME-fit index that will be used in this
paper is shown in Table 3.

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It seems to be difficult to decide on an adequate sample size for Benford. The consensus in the
literature [15] seems to be that a sample size of at least 50 to 100 might be required for Benford’d
law to be observed – if it does exist – but that a sample size of 500 or more might be preferred for
proper analysis.
Table 3: Classification of critical values
Decisions based on the chi-
square tests
Decisions based on the RMSE-
fit index
Decisions based on
the proportion tests
P-value
< 0.05
P-value
≥ 0.05
RMSE ≤ 0.02
Fit is very
good
P-value
< 0.05
P-value
≥ 0.05
Reject H0
Fit is poor
Cannot reject
H0
Fit is good
0.02
< RMSE ≤ 0.07
Fit is good
Reject H0
Fit is poor
Cannot reject
H0
Fit is good
0.07 < RMSE ≤
0.10
Fit is
acceptable
RMSE > 0.10
Fit is poor
Table 4 shows a set of possible guidelines for deciding whether or not a data set might be expected
to conform to Benford’s law [14-17].
Table 4: Suggested guidelines for data sets to conform to Benford’s law (or not)
Characteristics of data sets
conducive to the occurrence
of Benford’s law
Characteristics of data sets not
conducive to the occurrence
of Benford’s law
Systems/processes following a power law
Human interference and judgment
Multiplicative operations
Additive operations
Financial data
Natural lower and upper limits
Value span of multiple orders of magnitude
Data with a small value span
Distributions with positive skewness
Symmetrical distributions
Large sample size
Small sample size
Data independence
Autocorrelation (time series)
Data with dimensions
Assigned and ranking numbers
Numeric data
Data of different types or origins
Products of statistical distributions
Ordinal, repetitive, and classification data
2.4 The characteristics of Benford’s law
Figures don’t lie, but liars do figure. Possibly Mark Twain
The results obtained from applying a Benford test on a data set should be interpreted with care and
insight. The results from a Benford test should never be used as an absolute proof or disproof of the
presence of Benford’s law – nor, for example, the possible existence of data tampering. It can at
best be used to provide an indication of whether further investigation of the data set might be
appropriate. Special care should be taken in interpreting the results from typical statistical
goodness-of-fit tests. These tests are sensitive to small samples, and are usually designed either to
reject or not reject a hypothesis at a high level of confidence that might not be required for the
effective application of Benford analysis.
Traditionally, Benford’s law is applicable to data sets from natural and accounting origins. However,
there are some indications that it might also be applicable to data generated as the consequence of
mathematical operations (see Tables 5 and 6). For this reason, it is in part the purpose of this paper
to investigate the possibility that the data typically published as part of engineering and scientific
papers might also conform to Benford’s law.
Tables 5 and 6 shows the results from experiments performed by applying Benford analysis to a
selection of typical data sets. The purpose of these experiments is to demonstrate the
characteristics of Benford’s law, and to serve as motivation for some remarks about the
characteristics of Benford’s law. The second-to-last column shows the relative frequencies observed
from the data set in comparison with the relative frequencies of the Benford distribution. The last
column contains the final decision of the authors, based on the available evidence.

6
Data set A, the Fibonacci numbers, is a series with a high value for the first order auto-correlation
coefficient, which might indicate that the numbers are not independent, and that the series has no
dimensions, does not result from a natural process, and contains only additive operations; and yet
it is almost a perfect fit for Benford’s law. Data set A is a good example of an apparently inexplicable
exception to the suggested guidelines provided in Table 4.
Data set B, the prime numbers, and data set C, the square root, are both a poor fit to Benford’s
law; but, for no obvious reason, data set D, the factorials, does provide a good fit; and the same is
true for data set E, the power function. The good fit provided by data set E, the power function, is
important, since many natural systems and processes, such as the size of craters on the moon and
the height of solar flares, follow a power function. This might provide some reason that so many
data sets from natural processes tend to conform to Benford’s law.
The Benford distribution is discrete; but some continuous processes, such as exponential growth,
also conform to Benford’s law. As an example, data set F was obtained from the calculation of
compound interest, which does provide a good fit.
Further examples of the exceptions to Benford’s law arfe provided by data sets G and H. Values from
an exponential distribution provide a good fit, but not values from a normal distribution.
Data sets I, cost data, and J, population data, are typical examples of data sets that should conform
to Benford’s law. However, the fit for data set J is not very good. The reason for this might be the
fact that the available data does not contain populations of less than 1500, thus causing a lower
limit.
The exceptions to Benford’s law are difficult to explain without further research.
3 APPLICATIONS OF BENFORD’S LAW
The world looks neater from the precincts of MIT on the river Charles
than from the hurly-burly of Wall Street by the Hudson. Fischer Black.
Two typical possible applications of Benford’s law will be investigated and discussed in this section:
the Fisher-Mendel controversy, and papers selected from the South African Journal of Industrial
Engineering. The main purpose is to determine whether data extracted from these documents
conforms to Benford’s law; and the authors should therefore be absolved from any possible data
manipulation or data tampering.
3.1 The Fisher-Mendel controversy
Numbers don't lie, sir. Politics, poetry, promises - those are lies!
Numbers are as close as we get to the handwriting of God. Hermann Gottlieb
Gregor Johann Mendel gained posthumous recognition as the founder of the
modern science of genetics, primarily because of his paper, published in 1865, dealing with his
numerous experiments with peas [19]. However, Ronald Aylmer Fisher, probably one of the most
accomplished and respected statisticians of the 20th century, analysed Mendel’s data; and in a paper
published in 1936, he concluded that “the data of most, if not all, of the experiments have been
falsified so as to agree closely with Mendel’s expectations” [20].

7
Table 5 Summary results for the data sets investigated
E
Power function
2n
800
P-value = 1.00
Cannot Reject H0
Fit index = 0.001
Fit is very good
None
Yes
D
Factorial
n!
146
P-value = 0.16
Cannot Reject H0
Fit index = 0.025
Fit is good
None
Yes
C
Square
root
1000
P-value < 10-4
Reject H0
Fit index = 0.116
Fit is poor
2,4,5,6, 7,8 and 9
No
B
Prime
Numbers
1000
P-value < 10-4
Reject H0
Fit index = 0.063
Fit is acceptable
1,4,5,6,7,8 and 9
No
A
Fibonacci
numbers
1000
P-value = 1.0
Cannot reject H0
Fit index = 0.001
Fit is very good
None
Yes
Data set identification
Source of
data set
Sample size
Chi-square test
RMSE-fit
index test
Intervals rejected by
proportion tests
Graphical display of
observed (data set) and
expected (Benford’s
law)
relative frequencies for
the first digit
Conforms to Benford?

8
Table 6 Summary results for the datasets investigated
J
Population of South
African metropolitan
areas
219
P-Value < 10-4
Reject H0
Fit index = 0.021
Fit is acceptable
2
No
I
Cost data from
the Benford paper
741
P-Value = 0.05
Cannot Reject H0
Fit index = 0.015
Fit is very good
None
Yes
H
Numbers generated
from a normal
distribution
1000
P-Value < 10-4
Reject H0
Fit index = 0.138
Fit is poor
1,2,3,4,5, 6,8 and 9
No
G
Numbers generated from
an exponential
distribution
1000
P-Value = 0.41
Cannot Reject H0
Fit index= 0.011
Fit is very good
None
Yes
F
Compound interest
(Exponential growth)
100
P-Value = 0.94
Cannot Reject H0
Fit index= 0.008
Fit is very good
None
Yes
Data set
identification
Source of
Benford data set
Sample size
Chi-square test
RMSE-fit
index test
Intervals ejected
by proportion
tests
Graphical display
of
observed (data
set) and
expected
(Benford’s law)
relative
frequencies for
the first digit
Conforms to
Benford?
This accusation gave rise to a controversy, known as the Mendel-Fisher controversy, which in some
ways is still raging. Several attempts have been made to resolve the controversy [21]. A compromise
conclusion was reached, essentially saying that Fisher was probably correct from a purely statistical

9
point of view, although he might have been over-conscientious and conservative [21]. At the same
time, there is no conclusive evidence that Mendel – the scientist, Augustinian friar, and abbot of St
Thomas' Abbey – was guilty of data tampering. It should be mentioned that Fisher did not question
Mendel’s conclusions, but said only that “the data is too good to be true” [20,21], and admitted
that, if there were any data falsification, it might be due to an over-zealous assistant of Mendel who
might have been aware of what was expected, and possibly performed some selective sampling to
please the friar [21]. In the late 1950s, Fisher was also involved in another dispute, the so-called
cancer controversy [22], when he doubted that smoking cigarettes caused lung cancer, claiming that
his analysis did not provide conclusive proof of the existence of a relationship between smoking and
lung cancer. It has been suggested that, in this case, Fisher might have been guilty of selective
sampling [23]. It is conceivable that Fisher was not aware of Benford’s law when he wrote his paper
on the Mendel data, since Benford had published his paper only two years later. It seems
appropriate, therefore, to subject Mendel’s data to a Benford analysis.
For this purpose, Mendel’s original paper [19] was obtained and a data set extracted. The extraction
process involved some data filtering – for example, all of the data consisting of ratios was omitted.
Such a process of selective sampling should be performed with extreme care, since it can easily
introduce statistical bias. The results of this analysis are shown in Table 7 and Figure 6, and seem
to vindicate the already-mentioned conciliatory conclusions recently reached [20,21]. Regarding
Figure 6, the frequency of interval 5 seems too low and the frequency of interval 6 seems to be too
large. This might indicate selective or biased sampling, and could serve as the starting point of any
further investigation. Furthermore, intervals 5 and 6 contribute 63 per cent of the total chi-square
statistic, providing a possible reason that the chi-square test resulted in a rejection of the nul-
hypothesis.
Figure 6: Histogram of the first digit of the Mendel data in comparison with the Benford
histogram
Table 7: Statistical analysis of the Benford results for Mendel’s data
Number
1
2
3
4
5
6
7
8
9
Observed frequency of the first digit
30
30
19
12
2
15
11
7
5
Chi-square P-value = 0.0237
Reject H0
Sample
size
= 131
RMSE-fit Index = 0.0429
Fit is good
Test for proportions
P-value
0.20
0.26
0.62
0.88
0.06
0.12
0.37
0.93
0.77
Reject H0?
No
No
No
No
No
No
No
No
No
3.2 Benford analysis of papers selected from the South African Journal of Industrial
Engineering
Data is like garbage. You'd better know what you are going to do with it
before you collect it. Mark Twain
To investigate the use of Benford’s law to identify possible data tampering in papers published in
typical engineering journals, six papers from recent issues of the South African Journal of Industrial
Engineering were selected. These papers were not randomly selected, but rather because of the
amount of useable data they contained. The six papers were each subjected to a Benford analysis;
the results are summarised in Tables 8, 9, 10, 11, 12, and 13 and in Figures 7, 8, 9, 10, 11, and 12.

10
Figure 7: First digit relative frequencies
for Paper 1
Figure 8: First digit relative frequencies
for Paper 2
Table 8: Statistical analysis of the Benford results for Paper 1
Number
1
2
3
4
5
6
7
8
9
Observed frequency of the first digit
46
23
11
5
8
7
3
4
4
Chi-square P-value = 0.17
Cannot reject H0
Sample
size
= 111
RMSE-fit index = 0.045
Fit is good
Test for proportions
P-value =
0.07
0.54
0.56
0.19
0.84
0.91
0.32
0.61
0.73
Reject H0?
No
No
No
No
No
No
No
No
No
Table 9: Statistical analysis of the Benford results for Paper 2
Number
1
2
3
4
5
6
7
8
9
Observed frequency of the first digit
48
26
33
16
20
8
4
12
25
Chi-square P-value <10-4
Reject H0
Sample size = 192
RMSE-fit index = 0.042
Fit is good
Test for proportions
P-value =
0.12
0.14
0.05
0.52
0.20
0.16
0.03
0.48
<10-4
Reject H0?
No
No
No
No
No
No
Yes
No
Yes
The data set for Paper 1 passes all the tests, and therefore can be considered as conforming to
Benford’s law.
The data set for Paper 2 fails the chi-square and two of the proportion tests, but none of the other.
The proportion test for digit 9 indicates that this relative frequency might be an outlier, and might
be the reason that the data set fails the chi-square test, since this test is sensitive to outliers.
Further investigation showed that the data set for Paper 2 contains several probability values greater
than 0.9, which might be the cause of the outlier and thus the failure of the chi-square test. Since
it is known that probabilities do not necessarily conform to Benford’s law, these values can be
considered for removal from the data set; but this should be done with trepidation. Given the
available information and the preceding arguments, Paper 2 can be considered to conform to
Benford’s law.
Figure 9: First digit relative frequencies for
Paper 3
Figure 10: first digit relative frequencies for
Paper 4

11
Table 10: Statistical analysis of the Benford results for Paper 3
Number
1
2
3
4
5
6
7
8
9
Observed frequency of the first
digit
24
2
4
3
7
3
3
2
3
Chi-square P-value = 0.07
Cannot reject H0
Sample size =
51
RMSE-fit index = 0.078
Fit is acceptable
Test for proportions
P-value =
0.06
0.07
0.48
0.52
0.28
0.87
0.99
0.78
0.75
Reject H0?
No
No
No
No
No
No
No
No
No
Paper 3 seems to conform to Benford’s law in all respects.
The fit of Papers 4 and 5 is not very good, but might suffer from the same problem regarding outliers
as that discussed in the case of Paper 2; but it is still considered to conform to Benflord’s law.
Figure 11: First digit relative frequencies for
Paper 5
Figure 12: First digit relative frequencies for
Paper 6
Table 11: Statistical analysis of the Benford results for Paper 4
Number
1
2
3
4
5
6
7
8
9
Observed frequency of
the first digit
119
71
53
32
27
12
5
18
5
Chi-square P-value = 0.0002
Reject H0
Sample size
= 342
RMSE-fit index = 0.030
Fit is good
Test for proportions
P-value =
0.18
0.28
0.23
0.88
0.99
0.10
0.02
0.93
0.05
Reject H0?
No
No
No
No
No
No
Yes
No
No
Table 12: Statistical analysis of the Benford results for Paper 5
Number
1
2
3
4
5
6
7
8
9
Observed frequency
for the first digit
75
43
30
10
20
15
6
16
1
Chi-square P-value = 0.0043
Reject H0
Sample size
= 216
RMSE-fit index
= 0.031
Fit is good
Test for proportions
P=value =
0.14
0.38
0.54
0.01
0.47
0.88
0.06
0.13
0.00
Reject H0 ?
No
No
No
Yes
No
No
No
No
Yes
Table 13: Statistical analysis of the Benford results for Paper 6
Number
1
2
3
4
5
6
7
8
9
Observed frequency
for the first digit
27
19
16
15
17
8
14
9
6
Chi-square P-value = 0.05
Cannot Reject H0
Sample size
= 131
RMSE-fit index
= 0.042
Fit is good
Test for proportions
P-value =
1.67
0.66
0.07
0.48
1.52
0.19
1.69
0.64
0.00
Reject H0?
No
No
No
No
No
No
No
No
No

12
Paper 6 seems to conform to Benford’s law in all respects.
Furthermore, the graphical evidence points to a consistent tendency towards conformance with
Benford’s law for all the papers.
There are several reasons that typical data from engineering and scientific papers might not conform
to Benford’s law. This could include small sample sizes, upper limits, and limited data ranges for
variables such as probabilities, indexes and ratios, dependent data, data from different types,
sources and origins, etc. These factors might cause further investigation to be considered, although
it might not be necessary.
For the sake of transparency – and to heed the advice of Robert Louis Stevenson, among others:
“There is so much good in the worst of us, and so much bad in the best of us, that it hardly behooves
any of us to talk about the rest of us” – it should be revealed that the authors of this paper are the
authors of Paper 1, and that the main author of this paper is also the co-author of Paper 2.
Furthermore, it should be admitted that paper 6 is this very paper. The titles and authors of the
other papers will remain anonymous.
Given all the evidence, it is the authors’ considered opinion that, “on the balance of probabilities”,
the authors of the six papers investigated can be found “not guilty, beyond a reasonable doubt” of
any data tampering or unethical behaviour!
4 COMMENTS, CAVEATS, AND CONCLUSIONS
I often say that when you can measure what you are speaking about, and express it in numbers,
you know something about it; but when you cannot measure it, when you cannot express it in
numbers, your knowledge is of a meagre and unsatisfactory kind.
Lord Kelvin (William Thomson)
Some other interesting characteristics of Benford’s law have not been mentioned – for example [8]:
Benford’s law is related to Ziph’s law, known to linguists and used to study the frequency of words
in a manuscript. Benford’s law can be generalised beyond the first digit. However, the distribution
of the n-th digit, as n increases, rapidly approaches a uniform distribution. A Benford data set is
scale invariant – that is, it can be multiplied by a constant, and will still retain the Benford
characteristics. An extension of Benford's law can be used to predict the distribution of first digits
in other bases besides the decimal.
It has been stated that “the widely-known phenomenon called Benford’s law continues to defy
attempts at an easy derivation” [10]. In that sense, “most experts seem to agree that the ubiquity
of Benford’s law, especially in real-life data, remains mysterious” [10]. This characteristic of
Benford’s law complicates the decision about whether a data set should, or should not, conform to
Benford’s law. Benford’s law is by no means perfect – as is the case with most other statistical tests
– but it does provide another alternative, and a valuable way of performing certain kinds of statistical
analysis, when applicable.
Statistical inference is an invaluable tool for effective decision-making, but should be interpreted
with care. It should not be applied blindly, and some room should be left for the consideration of
good judgment, common sense, and even intuition based on experience and knowledge. In this
respect, the validity and value of graphical evidence, such as a graph of relative frequencies, should
not be under-estimated.
This paper has showed that Benford analysis can be applied to the investigation of the data published
as part of engineering or scientific papers. However, considering the implementation of such an
approach, similar to computerised testing for plagiarism, can be difficult to implement in practice.
Given the explosion in data availability, there is a need for effective scanning mechanisms to identify
the possible existence of aberrations and anomalies in large data sets. Benford’s law might be useful
in this kind of digital profiling. Furthermore, it seems as if the possible use of the results of a Benford
analysis to serve as a kind of process signature has not been investigated. This could be useful, for
example, in condition monitoring types of applications.

13
The available statistical and graphical evidence provides enough reason to declare that both Gregor
Johann Mendel and the authors of the selected papers published in the Journal should be exonerated
from any professional misconduct.
REFERENCES
There are more things in heaven and earth, Horatio,
than are dreamt of in your philosophy.
From Hamlet: Prince of Denmark by William Shakespeare
References to many publications dealing with Benford’s law are available [24]. However, for the
sake of brevity, only those publications that have been referenced or are the most significant or
informative to this paper are included in the list of references below.
[1] Weisstein, E.W. 2016. Hardy-Ramanujan number. MathWorld: A Wolfram Web Resource, available from
http://mathworld.wolfram.com/Hardy-RamanujanNumber.html [Accessed January 2017].
[2] Benford, F. 1938. The law of anomalous numbers. Proceedings of the American philosophical society,
pp.551-572.
[3] Dur tsch i, W. L. & Pacin i, C. 2004. The effective use of B en ford’ s law to assist in detecting fraud
in ac coun ting data. Journal of For en si c Accounting , (V ).
[4] Goldacre, B. 2011. Benford’s Law: Using stats to bust an entire nation for naughtiness. The Guardian,
Saturday 17 September 2011, available from http://www.badscience.net/2011/09/benfords [Accessed
January 2017].
[5] Hill, D.P. 1998. The first-digit phenomenon. American Scientist, July-August 1998.
[6] McGinty, J.C. 2014. Accountants increasingly use data analysis to catch fraud. The Wall Street Journal,
available from http://www.wsj.com/articles/accountants-increasingly-use-data-analysis-to-catch-fraud-
14 [Accessed January 2017].
[7] Wales, J. & Sanger, L. 2016. Benford’s Law. Wikimedia Foundation, Wikipedia Encyclopaedia, available
from https://en.wikipedia.org/wiki/Benford%27s_law [Accessed January 2017].
[8] Nigrini, M.J. 1999. I've got your number. Journal of Accountancy, May 1, 1999.
[9] Singleton, T.W. 2011. Understanding and applying Benford’s law. ISACA Journal, (3), available from
www.isaca.org/Journal/archives/2011/Volume-3/Pages/Understanding-and-Applying-Benfords-Law.aspx
[Accessed January 2017].
[10] Berger, A. and Hill, T.P. 2011. Benford’s law strikes back: no simple explanation in sight for mathematical
gem. The Mathematical Intelligencer, 33(1), pp.85-91.
[11] Hill, T.P. 1995. A statistical derivation of the significant-digit law. Statistical Science, (10).
[12] Newcomb, S. 1881. Note on the frequency of use of the different digits in natural numbers. American
Journal of Mathematics, (4).
[13] Montgomery, D.C. & Runger, G.C. 2011. Applied statistics and probability for engineers. John Wiley &
Sons.
[14] Cangur, S. & Ercan, I. 2015. Comparison of model fit indices used in structural equation modeling under
multivariate normality. Journal of Modern Applied Statistical Methods, (14).
[15] Kenny, D.A. 2015. Measuring model fit, available from http://davidakenny.net/cm/ fit.htm [Accessed
January 2017].
[16] Pike, D.P. 2008. Testing for the Benford property. School of Mathematical Sciences, Rochester Institute
of Technology, available from
www.researchgate.net/publication/251693892_Testing_for_the_Benford_Property, [Accessed January
2017].
[17] Tanaka, J.S. 1993. Some clarifications and recommendations on fit indices. Testing structural equation
models. Newbury Park, available from http://web.pdx.edu/~newsomj/semclass/ho_fit.doc [Accessed
January 2017].
[18] Bernstein, P.L. 1996. Against the gods: The remarkable story of risk. John Wiley & Sons.
[19] Mendel, G. 1865. Experiments in plant hybridization (1865). Available from
http://www.mendelweb.org/Mendel.html [Accessed January 2017].
[20] Franklin, A. 2008. Ending the Mendel-Fisher controversy. University of Pittsburgh Press.
[21] Pires, A.M. & Branco, J.A. 2010. A statistical model to explain the Mendel–Fisher controversy. Statistical
Science, (25).
[22] Fisher, R.A. 1958. Cigarettes, cancer and statistics. Centen Rev (2).
[23] Stolley, P.D. 1991. When genius errs: R.A. Fisher and the lung cancer controversy. American Journal of
Epidemiology, (133).
[24] Hürlimann, W. (ed.). 2006. Benford’s law from 1881 to 2006: A bibliography. Available from
arxiv.org/pdf/math [Accessed January 2017].