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South African Journal of Industrial Engineering August 2017 Vol 28(2), pp 1-13

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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.

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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].

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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?

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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

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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.

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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

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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].