麻豆传媒

ALEX had no idea what dark little secret he was about to uncover when he
asked his brother-in-law to help him out with his term project. As an
accountancy student at Saint Mary鈥檚 University in Halifax, Nova Scotia, Alex
needed some real-life commercial figures to work on, and his brother-in-law鈥檚
hardware store seemed the obvious place to get them.

Trawling through the year鈥檚 sales figures, Alex could find nothing obviously
strange about them. Still, he did what he was supposed to do for his project,
and performed a bizarre little ritual requested by his accountancy professor,
Mark Nigrini. He went through the sales figures and made a note of how many
started with the digit 1. It came out at 93 per cent. He handed it in and
thought no more about it.

Later, when Nigrini was marking the coursework, he took one look at that
figure and realised that an embarrassing situation was looming. His suspicions
hardened as he looked through the rest of Alex鈥檚 analysis of his
brother-in-law鈥檚 accounts. None of the sales figures began with the digits 2
through to 7, and there were just 4 beginning with the digit 8, and 21 with 9.
After a few more checks, Nigrini was in no doubt: Alex鈥檚 brother-in-law was a
fraudster, systematically cooking the books to avoid the attentions of bank
managers and tax inspectors.

It was a nice try. At first glance, the sales figures showed nothing very
suspicious, with none of the sudden leaps or dives that often attract the
attentions of the authorities. But that was just it: they were too regular. And
this is why they fell foul of that ritual he had asked Alex to perform.

Because what Nigrini knew鈥攁nd Alex鈥檚 brother-in-law clearly
didn鈥檛鈥攚as that the digits making up the shop鈥檚 sales figures should have
followed a mathematical rule discovered accidentally over 100 years ago. Known
as Benford鈥檚 law, it is a rule obeyed by a stunning variety of phenomena, from
stock market prices to census data to the heat capacities of chemicals. Even a
ragbag of figures extracted from newspapers will obey the law鈥檚 demands that
around 30 per cent of the numbers will start with a 1, 18 per cent with a 2,
right down to just 4.6 per cent starting with a 9.

It is a law so unexpected that at first many people simply refuse to believe
it can be true. Indeed, only in the past few years has a really solid
mathematical explanation of its existence emerged. But after years of being
regarded as a mathematical curiosity, Benford鈥檚 law is now being eyed by
everyone from tax inspectors to computer designers鈥攁ll of whom think it
could help them solve some tricky problems with astonishing ease. In two weeks鈥�
time, the US Institute of Internal Auditors will begin holding training courses
on how to apply Benford鈥檚 law in fraud investigations, hailing it as the biggest
advance in the field for years.

The story behind the law鈥檚 discovery is every bit as weird as the law itself.
In 1881, the American astronomer Simon Newcomb penned a note to the American
Journal of Mathematics about a strange quirk he鈥檇 noticed about books of
logarithms, then widely used by scientists performing calculations. The first
pages of such books seemed to get grubby much faster than the last ones.

The obvious explanation was perplexing. For some reason, people did more
calculations involving numbers starting with 1 than 8 and 9. Newcomb came up
with a little formula that matched the pattern of use pretty well: nature seems
to have a penchant for arranging numbers so that the proportion beginning with
the digit D is equal to log10 of 1 + (1/D) (see 鈥淗ere, there and
别惫别谤测飞丑别谤别鈥�).

With no very convincing argument for why the formula should work, Newcomb鈥檚
paper failed to arouse any interest, and the Grubby Pages Effect was forgotten
for over half a century. But in 1938, a physicist with the General Electric
Company in the US, Frank Benford, rediscovered the effect and came up with the
same law as Newcomb. But Benford went much further. Using more than 20 000
numbers culled from everything from listings of the drainage areas of rivers to
numbers appearing in old magazine articles, Benford showed that they all
followed the same basic law: around 30 per cent began with the digit 1, 18 per
cent with 2 and so on.

Like Newcomb, Benford did not have any really good explanation for the
existence of the law. Even so, the sheer wealth of evidence he provided to
demonstrate its reality and ubiquity has led to his name being linked with the
law ever since.

It was nearly a quarter of a century before anyone came up with a plausible
answer to the central question: why on earth should the law apply to so many
different sources of numbers? The first big step came in 1961 with some neat
lateral thinking by Roger Pinkham, a mathematician then at Rutgers University in
New Brunswick, New Jersey. Just suppose, said Pinkham, there really is a
universal law governing the digits of numbers that describe natural phenomena
such as the drainage areas of rivers and the properties of chemicals. Then any
such law must work regardless of what units are used. Even the inhabitants of
the Planet Zob, who measure area in grondekis, must find exactly the same
distribution of digits in drainage areas as we do, using hectares. But how is
this possible, if there are 87.331 hectares to the grondeki?

The answer, said Pinkham, lies in ensuring that the distribution of digits is
unaffected by changes of units. Suppose you know the drainage area in hectares
for a million different rivers. Translating each of these values into grondekis
will change the individual numbers, certainly. But overall, the distribution of
numbers would still have the same pattern as before. This is a property known as
鈥渟cale invariance鈥�.

Pinkham showed mathematically that Benford鈥檚 law is indeed scale-invariant.
Crucially, however, he also showed that Benford鈥檚 law is the only way to
distribute digits that has this property. In other words, any 鈥渓aw鈥� of digit
frequency with pretensions of universality has no choice but to be Benford鈥檚
law.

Pinkham鈥檚 work gave a major boost to the credibility of the law, and prompted
others to start taking it seriously and thinking up possible applications. But a
key question remained: just what kinds of numbers could be expected to follow
Benford鈥檚 law? Two rules of thumb quickly emerged. For a start, the sample of
numbers should be big enough to give the predicted proportions a chance to
assert themselves. Second, the numbers should be free of artificial limits, and
allowed to take pretty much any value they please. It is clearly pointless
expecting, say, the prices of 10 different types of beer to conform to Benford鈥檚
law. Not only is the sample too small, but鈥攎ore importantly鈥攖he
prices are forced to stay within a fixed, narrow range by market forces.

Random numbers

On the other hand, truly random numbers won鈥檛 conform to Benford鈥檚 law
either: the proportions of leading digits in such numbers are, by definition,
equal. Benford鈥檚 Law applies to numbers occupying the 鈥渕iddle ground鈥� between
the rigidly constrained and the utterly unfettered.

Precisely what this means remained a mystery until just three years ago, when
mathematician Theodore Hill of Georgia Institute of Technology in Atlanta
uncovered what appears to be the true origin of Benford鈥檚 law. It comes, he
realised, from the various ways that different kinds of measurements tend to
spread themselves. Ultimately, everything we can measure in the Universe is the
outcome of some process or other: the random jolts of atoms, say, or the
exigencies of genetics. Mathematicians have long known that the spread of values
for each of these follows some basic mathematical rule. The heights of bank
managers, say, follow the bell-shaped Gaussian curve, daily temperatures rise
and fall in a wave-like pattern, while the strength and frequency of earthquakes
are linked by a logarithmic law.

Now imagine grabbing random handfuls of data from a hotchpotch of such
distributions. Hill proved that as you grab ever more of such numbers, the
digits of these numbers will conform ever closer to a single, very specific law.
This law is a kind of ultimate distribution, the 鈥淒istribution of
Distributions鈥�. And he showed that its mathematical form is鈥�Benford鈥檚 Law.

Hill鈥檚 theorem, published in 1996, seems finally to explain the astonishing
ubiquity of Benford鈥檚 law. For while numbers describing some phenomena are under
the control of a single distribution such as the bell curve, many
more鈥攄escribing everything from census data to stock market
prices鈥攁re dictated by a random mix of all kinds of distributions. If
Hill鈥檚 theorem is correct, this means that the digits of these data should
follow Benford鈥檚 law. And, as Benford鈥檚 own monumental study and many others
have showed, they really do.

Mark Nigrini, Alex鈥檚 former project supervisor and now a professor of
accountancy at the Southern Methodist University, Dallas, sees Hill鈥檚 theorem as
a crucial breakthrough: 鈥淚t . . . helps explain why the significant-digit
phenomenon appears in so many contexts.鈥�

It has also helped Nigrini to convince others that Benford鈥檚 law is much more
than just a bit of mathematical frivolity. Over the past few years, Nigrini has
become the driving force behind a far from frivolous use of the law: fraud
detection.

In a ground-breaking doctoral thesis published in 1992, Nigrini showed that
many key features of accounts, from sales figures to expenses claims, follow
Benford鈥檚 law鈥攁nd that deviations from the law can be quickly detected
using standard statistical tests. Nigrini calls the fraud-busting technique
鈥渄igital analysis鈥�, and its successes are starting to attract interest in the
corporate world and beyond.

Some of the earliest cases鈥攊ncluding the sharp practices of Alex鈥檚
store-keeping brother-in-law鈥攅merged from student projects set up by
Nigrini. But soon he was using digital analysis to unmask much bigger frauds.
One recent case involved an American leisure and travel company with a
nationwide chain of motels. Using digital analysis, the company鈥檚 audit director
discovered something odd about the claims being made by the supervisor of the
company鈥檚 healthcare department. 鈥淭he first two digits of the healthcare
payments were checked for conformity to Benford鈥檚 law, and this revealed a spike
in numbers beginning with the digits `65鈥�,鈥� says Nigrini. 鈥淎n audit showed 13
fraudulent cheques for between $6500 and $6599鈥�related to
fraudulent heart surgery claims processed by the supervisor, with the cheque
ending up in her hands.鈥�

Benford鈥檚 law had caught the supervisor out, despite her best efforts to make
the claims look plausible. 鈥淪he carefully chose to make claims for employees at
motels with a higher than normal number of older employees,鈥� says Nigrini. 鈥淭he
analysis also uncovered other fraudulent claims worth around $1 million
in total.鈥�

Not surprisingly, big businesses and central governments are now also
starting to take Benford鈥檚 law seriously. 鈥淒igital analysis is being used by
listed companies, large private companies, professional firms and government
agencies in the US and Europe鈥攁nd by one of the world鈥檚 biggest audit
firms,鈥� says Nigrini.

Warning signs

The technique is also attracting interest from those hunting for other kinds
of fraud. At the International Institute for Drug Development in Brussels, Mark
Buyse and his colleagues believe Benford鈥檚 law could reveal suspicious data in
clinical trials, while a number of university researchers have contacted Nigrini
to find out if digital analysis could help reveal fraud in laboratory
notebooks.

Inevitably, the increasing use of digital analysis will lead to greater
awareness of its power by fraudsters. But according to Nigrini, that knowledge
won鈥檛 do them much good鈥攁part from warning them off: 鈥淭he problem for
fraudsters is that they have no idea what the whole picture looks like until all
the data are in,鈥� says Nigrini. 鈥淔rauds usually involve just a part of a data
set, but the fraudsters don鈥檛 know how that set will be analysed: by quarter,
say, or department, or by region. Ensuring the fraud always complies with
Benford鈥檚 Law is going to be tough鈥攁nd most fraudsters aren鈥檛 rocket
蝉肠颈别苍迟颈蝉迟蝉.鈥�

In any case, says Nigrini, there is more to Benford鈥檚 law than tracking down
fraudsters. Take the data explosion that threatens to overwhelm computer data
storage technology. Mathematician Peter Schatte at the Bergakademie Technical
University, Freiberg, has come up with rules that optimise computer data
storage, by allocating disk space according to the proportions dictated by
Benford鈥檚 law.

Ted Hill at Georgia Tech thinks that the ubiquity of Benford鈥檚 law could also
prove useful to those such as Treasury forecasters and demographers who need a
simple 鈥渞eality check鈥� for their mathematical models. 鈥淣igrini showed recently
that the populations of the 3000-plus counties in the US are very close to
Benford鈥檚 law,鈥� says Hill. 鈥淭hat suggests it could be a test for models which
predict future populations鈥攊f the figures predicted are not close to
Benford, then rethink the model.鈥�

Both Nigrini and Hill stress that Benford鈥檚 law is not a panacea for
fraud-busters or the world鈥檚 data-crunching ills. Deviations from the law鈥檚
predictions can be caused by nothing more nefarious than people rounding numbers
up or down, for example. And both accept that there is plenty of scope for
making a hash of applying it to real-life situations: 鈥淓very mathematical
theorem or statistical test can be misused鈥攖hat does not worry me,鈥� says
Hill.

But they share a sense that there are some really clever uses of Benford鈥檚
law still waiting to be dreamt up. Says Hill: 鈥淔or me the law is a prime example
of a mathematical idea which is a surprise to everyone鈥攅ven the
别虫辫别谤迟蝉.鈥�

Alex is not the real name of Nigrini鈥檚 former student

NATURE鈥橲 preferences for certain numbers and sequences has long fascinated
mathematicians. The so-called Golden Mean鈥� roughly equal to 1.62 and
supposedly giving the most aesthetically pleasing dimensions for
rectangles鈥攈as been found lurking in all kinds of places, from seashells
to knots, while the Fibonacci sequence鈥�1, 1, 2, 3, 5, 8 and so on, every
figure being the sum of its two predecessors鈥攃rops up everywhere in
nature, from the arrangement of leaves on plants to the pattern on pineapple
skins.

Benford鈥檚 law appears to be another fundamental feature of the mathematical
universe, with the proportion of numbers starting with the digit D given by
log10 of 1 + (1/D). In other words, around 100 脳 log2 (30 per cent) of such
numbers will begin with 鈥�1鈥�; 100 脳 log1.5 (17.6 per cent) with 鈥�2鈥�; down to 100
脳 log1.11 (4.6 per cent) with 鈥�9鈥�.

But the mathematics of Benford鈥檚 law goes further, predicting the proportion
of digits in the rest of the numbers as well. For example, the law predicts that
鈥�0鈥� is the most likely second digit鈥攁ccounting for around 12 per cent of
all second digits鈥攚hile 9 is the least likely, at 8.5 per cent.

Benford鈥檚 law thus suggests that the most common non-random numbers are those
starting with 鈥�10鈥︹��, which should be almost 10 times more abundant than the
least likely, which will be those starting 鈥�99鈥︹��.

As one might expect, Benford鈥檚 law predicts that the relative proportions of
1, 2, 3 and so on making up later digits of numbers become progressively more
even, tending towards precisely 10 per cent for the least significant digit of
every large number.

In a nice little twist, it turns out that the Fibonacci sequence, the Golden
Mean and Benford鈥檚 law are all linked. The ratio of successive terms in a
Fibonacci sequence tend toward the golden mean, while the digits of all the
numbers making up the Fibonacci sequence tend to conform to Benford鈥檚 law.