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Accounting numbers generally obey a mathematical law called Benford’s Law, and this outcome is so unexpected that manipulators of information generally fail to observe the law. Armed with this knowledge, it becomes possible to detect the occurrence of accounting data that are presented fraudulently. However, the law also allows for the possibility of detecting instances where data are presented containing errors. Given this backdrop, this paper uses data drawn from companies listed on the Johannesburg Stock Exchange to test the hypothesis that Benford’s Law can be used to identify false or fraudulent reporting of accounting data. The results support the argument that Benford’s Law can be used effectively to detect accounting error and fraud. Accordingly, the findings are of particular relevance to auditors, shareholders, financial analysts, investment managers, private investors and other users of publicly reported accounting data, such as the revenue services
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SAJEMS NS 9 (2006) No 3 341
AD Saville1
Gordon Institute of Business Science, University of Pretoria
Accounting numbers generally obey a mathematical law called Benford’s Law, and this outcome is
so unexpected that manipulators of information generally fail to observe the law. Armed with this
knowledge, it becomes possible to detect the occurrence of accounting data that are presented
fraudulently. However, the law also allows for the possibility of detecting instances where data are
presented containing errors. Given this backdrop, this paper uses data drawn from companies
listed on the Johannesburg Stock Exchange to test the hypothesis that Benford’s Law can be used
to identify false or fraudulent reporting of accounting data. The results support the argument that
Benford’s Law can be used effectively to detect accounting error and fraud. Accordingly, the findings
are of particular relevance to auditors, shareholders, financial analysts, investment managers, private
investors and other users of publicly reported accounting data, such as the revenue services.
JEL M 40
Albert Ein stein was playing his violin in
a duet with Werner Heisenberg, who was
accompanying him on the piano. After a while
Heisenberg slammed his hands down on the
keys and said: ‘It’s one, two, one, two, Einstein!
Can’t you count?’
(Arthur, 1993)
From the mid-1990s investment markets
witnessed a surge in the incidence of exposed
accounting frauds and irregularities which, in
turn, prompted a significant tightening in the
regulatory environment as part of a regulatory
effort to stamp out the occurrence of accounting
deceit.2 Although recent evidence suggests that
this regulatory response has been effective in
reducing the occurrence of dishonest accounting,
the impact has not been comprehensive.
Moreover, the experience of the past decade
demonstrates that, whilst the country, industry
and business detail behind the data distortions
vary, the cases share a common harmful ailment:
accounting frauds have resulted in considerable
destruction of investor wealth.3 In addition,
recent evidence shows that the number and
size of companies that are disclosing accounting
irregularities and frauds have grown with time.
For example, the number of restatements due
to accounting irregularities in the United States
(US) increased by over 150 percent between
1997 and 2001 (Floyd, 2003, 5). Moreover, the
median size of companies making restatements
in the US, measured by market capitalisation,
increased from $500 million in 1997 to $2 billion
in 2002 (Floyd, 2003, 7). In South Africa, the
trends have been similar, with a growing number
of firms reporting accounting irregularities and
frauds over the past decade.
Against this backdrop, and as noted, numerous
efforts are being made to improve accounting
standards and auditing practices. Regulators
are also at pains to make firms’ managers and
directors more sensitive to the consequences
of financial malpractice. However, the pace
342 SAJEMS NS 9 (2006) No 3
of progress is slow and the effects unfinished.
Moreover, human behaviour is such that
fraudulent practices will linger even in a world
of perfect accounting systems and watertight
auditing practices. Thus, those interested in the
accuracy of publicly reported accounting data
including auditors, shareholders, financial
analysts, investment managers, private investors
and other users of publicly reported accounting
data, such as the revenue services must remain
vigilant for fraudulent accounting practices.
Helpfully, at this juncture, a little known but
powerful mathematical law, called Benford’s
Law (Benford, 1938), presents itself as a
potentially potent tool for rooting out fraudulent
practices from a wide array of information sets,
include accounting data. Significantly, the law
has been used in a range of international settings
to detect data error and fraud, including the case
of accounting data. Despite this potential, it is
surprising to find that whilst Benford’s Law has
been used by practitioners in the South African
setting, no attempt has been made to publish
evidence on the effectiveness of Benford’s
Law in detecting accounting data error or
fraud in a domestic setting. This paper aims to
address this gap in research by exploring the
relevance of Benford’s Law in the detection of
anomalies in data presented by firms listed on
the Johannesburg Stock Exchange (JSE).
The remainder of the paper is divided into
five sections. Section 2 provides an overview
of Benford’s Law, while Section 3 examines
the mechanics of employing Benford’s Law to
detect accounting data irregularities as well as
the data set employed in this study. Section 4 is
devoted to analysing the results, and provides
comment on the reliability and relevance of the
tool as a detector of fraudulent or erroneous
accounting data. On this score, the findings of
this study suggest that Benford’s Law has the
capacity to play a helpful role in assisting users
of accounting data detect error or fraud in
financial information. These findings are in line
with expectations and concur with the results of
similar studies carried out in other countries.
Some comment is also made in this section on
areas for further research. Section 6 is devoted
to concluding remarks.
An overview of Benford’s Law
In 1881, the astronomer-mathematician
Simon Newcomb published a short article
in the American Journal of Mathematics’
describing his observation that books of
logarithms were more worn in the beginning and
progressively unspoiled throughout (Newcomb,
1881). From this, Newcomb inferred that
researchers (including fellow astronomers and
mathematicians, as well as biologists, sociologists
and other scholars) using the logarithmic tables
were looking up numbers starting with the digit 1
more often than numbers starting with the digit
2. Similarly, Newcomb inferred that researchers
were looking up numbers starting with the digit 2
more often than those beginning with the digit 3,
and so on (Hill, 1998: 1). After a short heuristic
argument, Newcomb (1881: 40) concluded
that the probability (P) that a number (D1) has
the first significant digit (that is, first non-zero
digit) d1 is:
P (D1 = d1) = log10 ,
 
 
 
where d1 {1, 2, … , 9}.
From Newcomb’s rule, it can be calculated that
the probability of 1 occurring as the first digit is
0.301 (or 30.1 percent). Similarly, the probably
of 2 being the first digit is 0.176 (17.6 percent).
In this vein, Table 1 shows the probabilities of
first digits based on the above equation. That
the digits are not equally likely comes as a
surprise to most observers. However, it is even
more striking that Newcomb (1881) was able to
claim the existence of an exact rule describing
the distribution of first digits.
Despite the profound insights offered,
Newcomb’s article went unnoticed. However,
more than half a century later, and independently
of Newcomb’s findings, American physicist Frank
Benford made exactly the same observation
about logarithmic books and concluded the
same first-digit law. But Benford went further
than Newcomb by testing his conjecture with
an ‘effort to collect data from as many fields as
possible and to include a wide variety of types
[of data]’ (Benford, 1938: 551). To be more
specific, Benford’s published findings were
SAJEMS NS 9 (2006) No 3 343
based on 20 229 observations from such diverse
data sets as areas of rivers, atomic weights and
street addresses (in all, 20 widely different data
sets were sampled). Benford’s findings indicated
that the data closely fitted the logarithmic law.4
Moreover, apart from this empirical advantage,
Benford’s paper benefited from a second factor:
it was published adjacent to a soon-to-be famous
physics paper. With Newcomb’s contribution
having become completely forgotten, the
logarithmic probability law came to be known
as Benford’s Law.
Table 1
Probabilities of first- and higher-order significant digits
Digit (= d) Probability of first
significant digit = d
Probability of
second significant
digit = d
Probability of third
significant digit = d
Probability of fourth
significant digit = d
0 Not Applicable 0.11968 0.10178 0.10018
1 0.30103 0.11389 0.10138 0.10014
2 0.17609 0.10882 0.10097 0.10010
3 0.12494 0.10433 0.10057 0.10006
4 0.09691 0.10031 0.10018 0.10002
5 0.07918 0.09668 0.09979 0.09998
6 0.06695 0.09337 0.09940 0.09994
7 0.05799 0.09035 0.09902 0.09990
8 0.05115 0.08757 0.09864 0.09986
9 0.04576 0.08500 0.09827 0.09982
Source: Nigrini (1999: 2)
Before proceeding, it is useful to offer an
intuitive explanation of Benford’s Law. Consider
making a deposit of R100 in a bank account
that pays interest at the rate of 10 percent per
annum. The first digit will continue to be 1
until the account balance rises to R200. This
will take a 100 percent increase which, at an
annual compound rate of 10 percent, would
take about 7.3 years. When the account balance
reaches R200, the first digit will be 2. However,
growing at 10 percent per annum, the account
balance will rise from R200 to R300 in about
4.2 years. Moving from R300 to R400 will take
about three years, and from R900 to R1 000
will require roughly 1.1 years. However, moving
from R1 000 (where the first digit is once again
1) to R2 000, will take 7.3 years. Thus earlier
digits have higher frequencies of occurrence,
with the law holding with any phenomenon that
has a constant or erratic growth rate (Nigrini,
1999: 2-3).
Interestingly, there is also a general significant-
digit law which includes first digits but also
higher order digits (which may be equal to 0)
(Hill, 1996).5 For example, the general law holds
that the probability that the second significant
digit (D2) of a number is equal to d2 is:
P (D2 = d2) =
 
 
 
where d2 {0, 1, … , 9}.
From this general law it follows that the second
significant digits, although monotonically
decreasing in frequency through the digits (as in
the case of first digits), are much more uniformly
distributed than the first digits. As noted, the
rule holds for higher order digits; to illustrate
this point, Table 1 shows the unconditional
probabilities of occurrence for the second, third
and fourth significant digits.
Furthermore, the general law also specifies
the joint distribution of significant digits. For
344 SAJEMS NS 9 (2006) No 3
instance, the general law allows for calculation
of the probability that the first and second digits
are 1 and 2, respectively. Importantly, the joint
distribution is not purely the probability of
the first digit multiplied by the probability of
the second digit. Rather, the significant digits
are dependent.6 To demonstrate this point, a
simple calculation shows that the unconditional
probability that the second digit is 2 is
But, the conditional probability that the second
digit is 2 given that the first digit is 1 is
(Hill, 1998: 2). As an aside, Benford’s Law is
the only probability distribution on significant
digits which is invariant under changes of scale
(for example, converting from English to metric
units or from Yen to Euros), or under changes
of base (for example, replacing base 10 by base
8 or base 2, in which case the logarithmic base
10 is replaced by logarithm to the new base)
(Hill, 1996).7
In proceeding, it is worth noting that in the
65 years since Benford’s article appeared there
have been numerous attempts to ‘prove’ the
law (Hill, 1998: 3). Indeed, by 1990 close on
100 papers had been published focusing on
explaining or deriving the law in theoretical
terms.8 But there have been two main stumbling
blocks to explaining the law. First, some data
sets satisfy the law, whilst others do not. Until
recently, there has not been a clear definition
of a general statistical experiment that would
predict which tables would comply with the
law. Second, although there was some success
in showing that Benford’s Law is the only set of
digital frequencies which remain fixed under
scale changes, none of the proofs were rigorous
as far as probability theory is concerned.
Recently, however, these stumbling blocks have
been removed by the discovery of mathematical
laws of probability which explain and predict
the appearance of the logarithmic distribution
(Hill, 1995a and 1996). In this vein, Hill (1996:
2) shows that if probability distributions are
selected at random, and random samples are
then taken from each of these distributions so
that the overall process is ‘unbiased’, then the
leading significant digits of the combined sample
will converge to Benford’s Law (Hill, 1996: 2).
More specifically, using modern mathematical
probability theory it has been shown that the
frequencies of significant digits will conform to
the law when data distributions are selected at
random and random samples are taken from
these distributions. As an aside, not all writers
are in agreement with Hill’s (1996) conclusion.
Brookes (2002), for instance, is critical of
Benford’s Law. However, Brookes (2002: 4)
acknowledges that in the case of data sets that
consist of ‘quantisized items such as oranges,
cows … trees [and money]’ these criticisms are
not serious.
Histrionics aside, the theorems alluded to
above explain why many tables of numerical data
follow the logarithmic distribution described by
Benford’s Law and why others do not. The latter
set includes items such as telephone numbers
in a given region that usually begin with the
same few digits, administered numbers such as
personal identify numbers, hourly wage rates,
bank account numbers, postal codes and tax
payer numbers. As already noted, however,
and significantly in the current argument,
the theorem also explains why a surprisingly
diverse collection of information tends to
obey Benford’s Law. Examples of such data
include large accounting tables, stock market
figures, tables of physical constants, numbers
appearing in newspaper articles, demographic
data, numerical computations in computing and
aspects of scientific calculations (Raimi, 1969;
Ashcraft, 1992; Dehaene and Mehler, 1992;
Hill, 1996 and 1998; Ley, 1996; Nigrini, 1999).
The explanation for conformity with Benford’s
Law now is well established: the data sets are
composed of samples from many different
Returning to the main focus of this paper, the
prevalence of the logarithmic distribution in
true accounting data sets has led to Benford’s
Law being used in an international setting to
detect fraud or fabrication of data in financial
documents under the hypothesis that when
people fabricate data they do not choose
numbers which follow a logarithmic distribution
(Hill, 1996). Moreover, it is well documented
that people cannot behave truly randomly even
when such behaviour is to their advantage
(Chapanis, 1953; Bakan, 1960; Neuringer,
1986; Hill, 1999). Further to this, recent studies
support the hypothesis that concocted data do
SAJEMS NS 9 (2006) No 3 345
not follow Benford’s Law closely. Nigrini (1996
and 1999) has led the way in this respect, by
amassing extensive empirical evidence of the
occurrence of Benford’s Law in many areas of
accounting data.
On the back of the accumulated evidence,
Nigrini has come to the conclusion that in a wide
variety of accounting situations, the significant-
digit frequencies of true data confirm closely to
Benford’s Law (see also Carslaw, 1988; Thomas
1989). Conversely, then, Benford’s Law serves
as an ideal tool for detecting variances between
true accounting data and data that have been
manipulated or that contains errors. However,
apart from providing a tool that can alert users
to possible errors or potential fraud, Benford’s
Law holds a second advantage over other
methods used to detect data corruption: the
law is easily applied (Nigrini, 1999: 1). Such a
tool for testing data conformity is described in
Section 3 below.
Application of Benford’s Law
3.1 Test method
The aim of the current study is to test the
potential effectiveness of Benford’s Law in
detecting data error or fraud in accounting
information produced by JSE-listed companies.
As a point of departure, it should be recognised
that testing need not be confined to the first-digit
level. Nigrini and Mittermaier (1997) provide a
review of the range of tests available. To start
with, because of the general law, testing can
be applied to higher-order digits as easily as to
first digits (Nigrini, 1999: 4). The law can also
be used to test joint frequencies, such as the
first-two, first-three or, more generally, first-n
digit combinations. Other tests are available.
For instance, the analyst can test for rounding
of numbers, which suggests estimation. Testing
for duplication of numbers or combinations
of numbers is also a potential investigative
tool that hints at fraudulent or administrative
manipulation. Thus, numbers can be binned
to test for conformity in various ways. Most
commonly, though, testing is done at the level
of first- or first-two significant digits. This paper
tests data conformity with Benford’s Law at
the level of the first-significant digit. This basis
for testing conforms to the broad-level testing
criteria established by Nigrini (2000).
Having identified the test level, the process
turns to establishing whether the observed
digit(s) deviate(s) significantly from the expected
frequencies derived from Benford’s Law. In
this regard, following Nigrini (2000) a simple
regression analysis is employed to assess the
significance of any observed deviations from the
expected frequencies.
Specifically, to test for conformity with
Benford’s Law, a regression line is estimated
of the form:
Yi = β0 + β1Xi + εi
where Yi is the value of the frequency of the
i-th significant digit(s) drawn from the sample
data; β0 and β1 are parameters; Xi is a known
constant, namely the value of the independent
variable (frequency of the ith significant digit[s])
as per Benford’s Law; and εi is a random error
term with mean E{εi} = 0 and variance σ2 =
{εi} = σ2; and εi and εj are uncorrelated so that
the covariance σij = 0 for all i, j where i ≠ j and
i = 1,2, … , n. A perfect correlation between the
sample data and Benford’s Law would yield:
β0 = 0; and
β1 = 1.
From this, a t-test is used to test the joint null
hypotheses that β0 = 0 and β1 = 1, which are
the necessary conditions for observed data to
conform to Benford’s Law.
Given the testing method, it becomes necessary
to establish the data sampling technique
adopted. Unfortunately, Benford (1938) offered
no comment in this regard. Indeed, some writers
have gone so far as to hint at Benford having
mined the data analysed (Scott and Fasli, 2001:
7).9 Elsewhere, little insight is offered into
suitable data sampling techniques. For this
reason, this paper adopts a more ‘classical’
sampling stance by observing principles that are
widely recognised as the basis for generating
adequate samples: the samples used are random
and sufficiently large and variable to deliver
test statistics that offer an appropriate level of
precision. The data set is described below.
346 SAJEMS NS 9 (2006) No 3
3.2 Data set
To test the potential of Benford’s Law to detect
error or fraud in accounting data, two data sets
are employed. The first consists of a sample of
‘errant’ companies that were listed on the JSE
during the five-year period 1 July 1998 to 30 June
2003. These companies are commonly suspected
or known to have committed accounting fraud
or produced erroneous data, and their shares
were either suspended or delisted during the
reference period as a consequence.10 This
sample of 17 so-called ‘errant’ companies is
detailed in Table 2. One firm, Amalia Gold
Mining and Exploration Company Limited, was
dropped from the sample due to lack of data.
Table 2
Errant companies (1 July 1998-30 June 2003)
Company name Date of suspension or delisting
Amlac Limited 6 May 2002
Beige Holdings Limited 27 September 1999*
Essential Beverage Holdings Limited 1 July 2002
Internet Gaming Corporation Limited 4 November 2002
Leisurenet Limited 6 October 2000
Macmed Limited 2 July 2001
Noble Minerals Limited 1 July 2002
Oxbridge Online Limited 1 July 2002
REF Finance and Investment Corporation Limited 8 January 2002
Regal Treasury Bank Holdings Limited 27 June 2001
Shawcell Telecommunications Limited 18 January 2002
Taufin Holdings Limited 2 June 2003
Terrafin Limited 24 June 2002
Tigon Limited 18 January 2002
Tridelta Magnet Technology Holdings Limited 27 August 2001
Unifer Holdings Limited 19 June 2002
Whetstone Industrial Holdings Limited 19 April 2001
* Beige Holdings Limited’s suspension was subsequently lifted by the JSE.
Source: Alexander and Oldert (2003)
In order to verify the effectiveness of the above
test of Benford’s Law, data drawn from a control
group of an equal number (17) of companies
was used to test for ‘false positives’. This second
sample consists of a group of firms, as ranked
by Ernst and Young (2002), as having the top
reporting standards amongst listed companies
on the JSE.11 The Ernest and Young survey is
generated annually. For the sake of the tests
conducted in this study, in-sample period data
were drawn from the results of the 2002 survey.
This was done to ensure that the data sets used
are homogenous. It is also believed that using
the 2002 data set allows for sufficient time to
have elapsed from the date of the survey for any
data anomalies to have been reported or to have
emerged. This sample of so-called ‘compliant’
firms is detailed in Table 3.
SAJEMS NS 9 (2006) No 3 347
Table 3
‘Compliant’ companies (2002)
Company name Company name (continued)
ABSA Group Limited Illovo Sugar Limited
African Bank Investments Limited Kersaf Investments Limited
African Oxygen Limited Liberty Group Limited
Allan Gray Property Trust Nampak Limited
AngloGold Limited Nedcor Limited
Aveng Limited Pretoria Portland Cement Company Limited
Anglovaal Mining Limited Sanlam Limited
Firstrand Limited Sasol Limited
Gold Fields Limited
Source: Adapted from Ernst and Young (2002)
to explore statements that are more likely to
include errant data. The most obvious place to
search for data error is in the income statement.
Thus, testing is done on first-digit data drawn
from the income statement. The other principal
statements produced by firms in their annual
financial reports – namely the cash flow, change
of equity and balance sheet statements are less
prone to manipulation. That said, data error
or fraud that arises in the income statement
is likely to percolate into derived statements
that include statements of change in equity and
balance sheets. So, to eliminate the potential for
double-counting of errors, the data set is based
on income statement data.
Third, in the case of errant firms, only the
last set of publicly reported information is
used. For ‘compliant’ companies, the sample
set is drawn from the 2002 financial year, as
explained above.
Thus, two sets of data are produced by
the sampling method, namely: (a) income
statement first-digit data drawn from ‘errant
companies on a per company basis and (b)
income statement first-digit data drawn from
‘compliant’ companies on a per company basis,
with the income statement data consisting of
30 line items as reported by the companies.
Accordingly, the full data set consists of 1 020
income statement observations as reported
Data drawn from the financial statements of
the two sets of companies are confined by
three additional parameters. First, the tests
run are confined to raw data whose significant
number frequencies are expected to follow a
geometric sequence when ordered and counted.
Raw accounting data read as line items are
appropriate for testing. Numbers that are a
function of more than one set of other numbers
(such as earnings per share, which is a function
of earnings and the number of share in issue)
are not expected to follow Benford’s Law.12 To
ensure data homogeneity, the same line items
are used for all companies, as published by
data vendor I-Net Bridge’s Financial Analysis
System (FAS). Moreover, the data that are
sampled are ‘as reported’, which thus excludes
all possible influences of adjustments that are
typically made by data vendors in their efforts
to standardise accounting data. In proceeding,
it should be noted that the raw data identified
satisfy the main criteria for having expected
digit frequencies that are Benford-like, namely:
the numbers describe the sizes of similar
phenomena; the numbers have no built-in
maximums or minimums; and the numbers are
not assigned numbers (such as bank account
numbers) (Nigrini and Mittermaier, 1997).
Second, because the aim of the tests is to
identify data manipulation, it makes sense
348 SAJEMS NS 9 (2006) No 3
by the companies. The sampling method then
binned data on a per company basis, with testing
at the company level justified by the argument
that knowing that a group of companies employ
errant or questionable reporting practices is of
marginal use when compared to the knowledge
that a single company adopts such reporting
Thus, for each company the reported income
statement data are binned. The binned data
frequencies are then regressed on theoretical
frequencies to test for significant deviations from
Benford’s Law. It is expected that the testing
process would reveal significant deviations from
Benford’s Law in the case of ‘errant’ companies,
whilst the frequencies generated by ‘compliant’
company data are expected to observe Benford’s
Law. In proceeding, it ought to be noted that in
a priori testing, rejection of the null hypothesis
does not prove data error , bias or fraud
legitimate explanations for deviations are
sometimes found. Rather, a positive test result
signals potential data problems, which the data
user should then employ as grounds for a more
detailed examination of the information. This
argument, however, does not necessarily apply
in the case of backward-looking tests. Related
to this point, it must be recognised that the
unit of analysis is the firm, although clearly it
is not firms that falsify data, but rather agents
of the firm. However, detection of data error
at the firm level is arguably a first, necessary
step required in any search for the existence of
fraudulent company data (this point is returned
to below).
Test outputs
4.1 Test results
Tables 4 and 5 set out the test results on a per
company basis. Table 4 deals with the results
of tests conducted on ‘errant’ companies, and
shows the estimated values of β0 and β1; the
standard deviation of the estimated values; and
the t-statistic on the estimated values.
The acceptance of the independent null
hypotheses that β0 = 0 and β1 = 1 at the
five percent level of significance is indicated by
an asterisk on cell entries in Table 4. However,
to satisfy the test requirements, it is necessary
that β0 = 0 and β1 = 1 lie within two standard
deviations of the estimated values of β0 and β1.
Accordingly, the test results lead to acceptance of
the null hypothesis that β0 = 0 in 13 of 17 cases.
However, as can be inferred from the estimates
of β1, in all 13 cases the test results reject the null
hypothesis that β1 = 1 at the five percent level.
Hence, the joint requirement that β0 = 0 and β1
= 1 is rejected in all of these cases. As an aside,
there are three instances of significant estimates
of β1. But all three results fail to meet the criteria
of β1 = 1 lying within two standard deviations
of the estimated value of β1. Moreover, none of
these three cases coincide with acceptance of
the null hypothesis that β0 = 0. Further to this,
it is interesting to note that four estimates of β1
carry the wrong sign. These cases hint at ‘extreme’
violation of Benford’s Law: as first-digits increase
from one through to nine, the frequency of first-
Table 4
Test results on errant company data sets
Company Estimate
of β0
of β1
Amlac –0.11 0.03 –3.75 2.01* 0.22 9.11
Beige 0.17 0.06 3.10 –0.54 0.41 –1.32
Essential 0.13* 0.15 0.89 –0.19 1.10 –0.17
Igaming 0.08* 0.21 0.40 0.25 1.53 0.17
Leisurenet –0.12 0.05 –2.35 2.07* 0.37 5.55
Macmed 0.09* 0.06 1.42 0.19 0.47 0.40
SAJEMS NS 9 (2006) No 3 349
Noble –0.28 0.10 –2.66 3.50* 0.77 4.54
Oxbridge 0.11* 0.11 0.96 0.01 0.84 0.02
Refcorp 0.08* 0.09 0.90 0.28 0.65 0.43
Regal 0.13* 0.09 1.46 –0.18 0.66 –0.27
Shawcell 0.16* 0.13 1.18 –0.41 0.98 –0.42
Taufin 0.03* 0.06 0.56 0.70 0.44 1.59
Terrafin 0.02* 0.14 0.17 0.79 1.02 0.78
Tigon 0.08* 0.13 0.59 0.31 0.97 0.32
Tridelta 0.10* 0.16 0.63 0.08 1.20 0.07
Unifer 0.00* 0.07 0.04 0.98 0.49 2.00
Whetstone –0.02* 0.09 –0.23 1.18 0.66 2.00
Sample 0.04 0.10 0.19 0.65 0.75 1.46
‘early’ data manipulation has a cascading effect. To
put the argument differently, misstatement of line
items that occur low down in the income statement
would mean that a random sample of first digits
may comply with Benford’s Law due to the possible
compliance of earlier numbers which, in the case
of ‘late’ manipulation would make up the majority
of first digits. Thus, from the findings presented
in Table 4 it is inferred that it is more likely that
data manipulation in the current sample occurred
early in the income statement rather than late in the
statement. This, then, sharpens the fraud detection
tool as it is not the company that perpetrates a
fraud, but rather agents of the company and, given
the above arguments, most likely agents that are
able to influence ‘early’ line items. However, as
noted, the unit of analysis in the current study is
the firm, and so a more detailed study is left for
investigation elsewhere.
These comments aside, continuing with the
argument, whilst it may be useful to know that
‘errant’ companies fail to comply with Benford’s
Law, the test only becomes a useful screening
tool if it can be shown that ‘compliant’ companies
generate first-digit frequencies that conform to
Benford’s Law. Consequently, the second set of
tests ensures against Type II error. Given this
backdrop, the test results for the 17 ‘compliant’
companies are reported in Table 5, which sets
digits increases. Accordingly, first-digit
distributions in these data sets are highly suspect.
That aside, and in short, none of the data sets
tested passes the test conditions established for
conformation to Benford’s Law.
Thus, the preliminary finding, based on the
above sample set, is that Benford’s Law is a
useful indicator of the existence of fraudulent
or erroneous data. All 17 companies that are
believed or found to have generated fraudulent
data over the sample period fail the test of
conformity of the distribution of first significant-
digits with Benford’s Law. It is unsurprising to
note that the estimated values based on pooled
data for the 17 ‘errant’ companies indicates that,
if measured as a group, the first significant digit
frequencies fail to conform to Benford’s Law.
As an aside, in the case of ‘errant’ companies it
is evident that the non-compliance of the data with
Benford’s Law can occur due to manipulation of
line items at any level of the income statement.
However, that all companies fail to satisfy the
intercept and slope aspects of the test implies that
data manipulation in the sample occurs in line
items that appear close to the top of the income
statement (the overstatement of revenue is the most
obvious culprit). More to the point, the higher up
the statement that manipulation occurs, the greater
the deviance of the balance of the statement as the
350 SAJEMS NS 9 (2006) No 3
out the estimated values of β0 and β1; the standard
deviation of the estimated values; and the t-statistics
on the estimated values.
is not found to be significantly different from
zero at the five percent level (although the
estimate is significant at the 10 percent level).
Importantly, of the 16 estimates of β1 that are
found to be significantly different from 0, only
three estimates fail to meet the further condition
that β1 = 1 lies within two standard deviations
of the estimated value of β1. Thus, of the set of
‘compliant’ companies, 13 of the 17 firms pass
the joint test of β0 = 0 and β1 = 1, indicating
conformity with Benford’s Law. It is interesting
to note that the estimated values based on
Table 5
Test results on ‘compliant’ company data sets
Company Estimate
of β0
of β1
ABSA –0.01* 0.03 –0.31 0.96* 0.21 4.50
ABIL –0.01* 0.05 –0.22 1.09* 0.35 3.11
Afrox –0.01* 0.03 –0.40 1.12* 0.25 4.42
Allan Gray –0.07* 0.04 –1.83 1.64* 0.29 5.71
Anglogold 0.04* 0.03 1.12 0.65* 0.25 2.57
Aveng –0.03* 0.06 0.06 1.31* 0.45 2.93
Avmin –0.01* 0.05 –0.16 1.07* 0.39 2.77
Firstrand 0.01* 0.05 0.32 0.87* 0.35 2.51
Goldfield 0.04* 0.04 0.91 0.64 0.32 1.96
Illovo 0.02* 0.04 0.40 0.85* 0.31 2.71
Kersaf 0.01* 0.05 0.12 0.94* 0.39 2.41
Liberty –0.07* 0.04 –1.85 1.59* 0.26 6.09
Nampak –0.05* 0.03 –1.44 1.43* 0.24 5.87
Nedcor –0.04* 0.02 –2.13 1.36* 0.14 9.91
PPC 0.05* 0.03 1.64 0.55* 0.23 2.41
Sanlam –0.04* 0.03 –1.22 1.35* 0.24 5.71
Sasol 0.03* 0.04 0.62 0.77* 0.31 2.48
Sample –0.01 0.04 –0.26 0.65 0.75 1.46
The results set out in Table 5 for the sample of
‘compliant’ companies indicate that the null
hypothesis that β0 = 0 cannot be rejected at
the five percent level for any of the companies.
Moreover, in all 17 cases, β0 = 0 lies within two
standard deviations of the estimated values of β0.
Thus, all 17 of the ‘compliant’ companies have
significant first-digit frequencies that indicate
conformity with Benford’s Law in the case of
β0 = 0. In considering the estimates of β1, the
coefficient is significant in 16 of the 17 cases.
The estimate of β1 on Goldfields (β1 = 0.64)
SAJEMS NS 9 (2006) No 3 351
pooled data for the 17 ‘compliant’ companies
indicates that the group’s first significant-digit
frequencies conform to Benford’s Law, with
β0 = 0 and β1 = 1 for the group.
As a final comment on the estimated values
of β0 and β1, it is noteworthy that the standard
errors on the estimates in the case of ‘errant’
companies (0.10 and 0.75, respectively) are
more than twice the size of standard errors on
the estimates β0 and β1 in the case of ‘compliant’
companies. This result offers further anecdotal
evidence of the superior ‘quality’ of ‘compliant’
company data over ‘errant’ company data.
4.2 Implications and limitations
In short, the results of the testing procedure
indicate that conformity to Benford’s Law may
serve as a robust tool forewarning users of
accounting data of the potential existence of
data error or fraud. The results are particularly
encouraging in this regard in that the test
procedure yielded a false-positive result in
four of 34 cases (11.8 percent of the sample).
Put differently, when applied at the time of
annual financial reporting to the above sample
of ‘errant’ and ‘compliant’ companies, the test of
Benford’s Law correctly identified 88.20 per cent
of the cases (30 of 34 companies), and correctly
identified 100.0 per cent of ‘errant’ cases.13 The
reason for this appears to be elegantly simple:
like supernovae, fraudulent companies give
themselves away by shining more brightly than
their peers as they zealously thrash away their
final moments.
Nevertheless, whilst these early results of the
application of Benford’s Law yield encouraging
findings, the test procedure and data set have
limitations that suggest further research is
required. Some of the more obvious limitations
are identified below.
First, the data collection method may include
an obvious source of sample bias in that with
the benefit of hindsight, the status of ‘errant’
and ‘compliant’ companies was known before
testing was conducted. This begs the question
of whether the test method would be as reliable
in the case of live data, that is, as a prediction
tool (where the value of the instrument is
unambiguously greatest). There is no cause to
doubt that this is the case. Nevertheless, testing
of live data would go some way in confirming
the tool’s validity.
Second, and related to this point, the results
reveal that the test functions in a highly
effective fashion in the tails of the distribution
correctly failing ‘errant’ companies and
passing ‘compliant’ companies. However, the
data set used in this study offers no insight as
to ‘what goes on in between’. Over most of the
sample period there were in excess of 500 listed
companies on the JSE. Thus, this study covers
less than 10 per cent of the population. A broader
study is required to establish the effectiveness
of the tool across all firms. Until such time,
then, the instrument is arguably best used as an
indicator of potential data error or fraud rather
than a corroborator of data problems.
Third, the results offer no guide as to whether
all companies that fail the test ultimately fail
and, if so, what the extent of the lag in time is
between detection and failure.
Fourth, in the international setting, Benford’s
Law has been applied more widely than
accounting data as the basis for detecting
data error or fraud. Indeed, the potential
applications of the law are wide. For instance,
the law has been identified as relevant to the
interrogation of design efficiency (Hamming,
1970 and Knuth, 1981 in Scott and Fasli, 2001),
the examination of authenticity of mathematical
models (Varian, 1972 in Scott and Fasli; Nigrini,
1996), assessment of the validity of research
results (Matthews, 1999: 26) and the examination
of data storage and data management efficiency
(Nigrini, 1999). Moreover, as noted in Section
2, the tool also is applicable as an instrument
for detecting fraud in claims (such as insurance
claims and expense account claims), payments
(bank payments and payroll disbursements)
and tax fraud (income declarations and expense
claims). However, constraints of time confine
the extant study to a consideration of accounting
data problems amongst listed firms. Broader,
and more detailed, studies of Benford’s Law
should address these limitations.
352 SAJEMS NS 9 (2006) No 3
Over the past decade, the frequency of
accounting data error and fraud has increased
in the international and domestic settings.
The adverse economic effects of these data
problems are considered to be material. For
this reason, broad-based efforts are being made
by the accounting and auditing professions and
regulatory authorities to reduce the incidence
of data error and fraud. However, even in a
world where recording and reporting of data
is potentially error free, elements of human
behaviour (such as greed and deceit) will linger
on, causing data error and fraud to persist.
Moreover, the pace at which progress in
accounting, auditing and regulatory advances
are being made is slow. For these reasons, error
and fraud detection instruments are likely to
remain important instruments in the toolkits
of auditors, shareholders, financial analysts,
investment managers, private investors and other
users of publicly reported accounting data, such
as the revenue services. One such potential tool
is Benford’s Law. However, whilst the potential
effectiveness of the law has been established
in the international literature, the domestic
research environment is silent on the topic.
Accordingly, this paper examines the potential
effectiveness of Benford’s Law in the detection
of data error and fraud in a South African
setting. To examine the case, a simple regression
tool is applied to data generated by a set of 34
companies listed on the JSE. For the sake of
the study, the test sample consists of data drawn
from an equal number of so-called ‘errant’ and
‘compliant’ companies. The results of the study
are convincing, with the tool correctly failing
all 17 of the ‘errant’ companies; three of the
17 ‘compliant’ companies fail the test. Despite
the incidence of false–positive results, the
number is considered to be sufficiently small
(11.2 percent of the full sample) to conclude
that Benford’s Law has the capacity to serve
as an effective indicator of data problems in
accounting information. Moreover, under test
conditions that are broader than the a priori
conditions that were set, the success rate of
the test climbs to 97.1 percent. Further, whilst
the study has some limitations, none of these is
considered to be sufficient to challenge the basic
result: Benford’s Law has the potential to act as
a highly effective detector of data error or fraud
in accounting information.
1 The author would like to thank Kerry Hadfield,
Jim Harris, Warwick Lucas, Zane Spindler, Hunter
Thyne and John Verster for useful contributions
made to this paper; the author also acknowledges
the helpful comments provided by two anonymous
editors. However, the usual caveats apply.
2 International examples embrace a diverse set
of high-profile companies that includes Enron,
WorldCom, Lucent, Adelphia, Ahold, Tyco, Intel,
AOL-Time Warner and Global Crossing. As with
the international environment, the South African
business environment is scattered with examples
of accounting frauds and irregularities. Some
companies that have engaged such practices are
listed in Section 3 of this paper.
3 D’Agostino and Williams (2002) identify 919
cases of accounting restatements made by listed
companies in the United States (US) between
1 January 1997 and 30 June 2002. The study
finds that losses in market capitalisation of
US$100 billion occurred over the reference period.
See also Floyd (2003), where comment is made on
the growing incidence of accounting irregularities
amongst listed firms.
4 It ought to be noted that the validity of Benford’s
(1938) findings has been drawn into question by
some researchers. For example, Scott and Fasli
(2001: 5) note that Benford’s claim that the tested
data sets conformed to his law rested entirely
on the apparent similarity of the numbers. To
be sure, Benford made no attempt to test the
goodness of fit of the data. However, this has not
led to the rejection of Benford’s Law. Rather,
this shortcoming in Benford’s work has led to the
refinement of our understanding of the types of
data to which the law applies (Scott and Fasli,
2001: 2).
5 In his paper, Newcomb (1881) also determined the
probability of the ten second digits, independent of
the first digits (Brookes, 2002: 1).
6 Hill (1995a) provides the exact formulas of the
joint probability calculations.
7 Pinkham (1961) provided a key development in the
understanding of Benford’s Law by arguing that
SAJEMS NS 9 (2006) No 3 353
for any digit-distribution law to hold consistently, it
would have to be scale invariant. Pinkham’s (1961)
proof was later extended by Hill (1995b).
8 See Raimi (1976) for an early review of the
literature and Scott and Fasli (2002) for a more
recent literature survey. Three main groups of
explanations emerge from these literature surveys.
The first set argues that Benford’s Law is due to
the numbering system that we use to count upward
through the natural numbers. The second group of
mathematical explanations is based on the notion
of ‘randomness’ and the central limit theorem.
The third approach to deriving Benford’s Law is
termed ‘ontological’ because it asks: ‘What form
would a digit law take if such a law existed?’ Scott
and Fasli (2001: 3-5) and Brookes (2002) offer
comment in this regard. That aside, of these three
approaches, the second remains the most widely
accepted plausible explanation for conformity of
a data set to Benford’s Law (Scott and Fasli, 2001:
9 It is noteworthy that the test statistics generated by
Scott and Fasli (2001: 6) to interrogate Benford’s
(1938) results conform to Benford’s Law.
10 The companies identified by the author that are
commonly suspected or believed to have published
false or fraudulent data were supplied by a
group of ten investment brokers and managers
representing five different financial services firms
who dealt in listed companies over the reference
11 It is acknowledged that Ernst and Young’s
‘Excellence in Financial Reporting’ is not intended
by the authors to test or validate the authenticity
(correctness) of the numbers reported in financial
statements. Rather, in the absence of such a tool,
the report is used here as a proxy for indicating the
authenticity of reported accounting data.
12 To illustrate this point, all ending digits in earnings
per share figures are expected to be distributed
with equal probability. Further, first digit counts on
financial ratios, such as return on equity or return
on assets are, in many instances, likely to conform
more closely to a binary distribution than to the
distribution implied by Benford’s Law.
13 It is interesting to note that at the 10.0 per cent
level of significance and allowing for true values
of βi to lie within three standard deviations of the
estimated βi values all of the 17 ‘errant’ companies
continue to fail the test, whilst the number of false-
positive results declines to one. Thus, under this
set of broader test criteria, the overall success rate
of the test climbs to 97.1 per cent.
1 ALEXANDER, E. & OLDERT, N. (eds.) (2003)
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6 BROOKES, D. (2002) “Naked-eye quantum
mechanics: Practical applications of Benford’s law
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7 CARSLAW, C. (1988) “Anomalies in income
numbers: Evidence of goal oriented behavior”,
Accounting Review, 63: 321-327.
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behavior”, American Psychologist, 8: 332.
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Chairman, Committee on Banking, Housing, and
Urban Affairs. United States General Accounting
Office: Washington.
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linguistic regularities in the frequency of number
words”, Cognition, 43: 1-29.
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354 SAJEMS NS 9 (2006) No 3
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