[OldBits] Benford's Law

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From: Rohit Khare (Rohit@KnowNow.com)
Date: Mon Oct 09 2000 - 13:39:31 PDT

[Google: "We may not be doing much for b2b ecommerce, but we're a
powehouse in bar2bar betting!" :-) RK]


[Archive: 10 July 1999]
The power of one

Everyday numbers obey a law so unexpected it is hard to believe it's
true. Armed with this knowledge, says Robert Matthews, it's easy to
catch those who have been faking research results or cooking the books

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's University in Halifax, Nova
Scotia, Alex [not the student's real name] needed some real-life
commercial figures to work on, and his brother-in-law's hardware
store seemed the obvious place to get them.

Trawling through the year's 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's
analysis of his brother-in-law's 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's brother-in-law was a fraudster, systematically
cooking the books to avoid the attentions of bank managers and tax

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--and Alex's brother-in-law clearly
didn't--was that the digits making up the shop's sales figures should
have followed a mathematical rule discovered accidentally over 100
years ago. Known as Benford's 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's 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's
law is now being eyed by everyone from tax inspectors to computer
designers--all of whom think it could help them solve some tricky
problems with astonishing ease. In two weeks' time, the US Institute
of Internal Auditors will begin holding training courses on how to
apply Benford's law in fraud investigations, hailing it as the
biggest advance in the field for years.

The story behind the law's 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'd
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 "Here, there and everywhere").

With no very convincing argument for why the formula should work,
Newcomb's 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
"scale invariance".

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

Pinkham's 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's 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's
law. Not only is the sample too small, but--more importantly--the
prices are forced to stay within a fixed, narrow range by market

Random numbers

On the other hand, truly random numbers won't conform to Benford's
law either: the proportions of leading digits in such numbers are, by
definition, equal. Benford's Law applies to numbers occupying the
"middle ground" between the rigidly constrained and the utterly

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's 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 "Distribution of Distributions". And he showed that
its mathematical form is...Benford's Law.

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

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

It has also helped Nigrini to convince others that Benford's 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's law--and that deviations from the
law can be quickly detected using standard statistical tests. Nigrini
calls the fraud-busting technique "digital analysis", and its
successes are starting to attract interest in the corporate world and

Some of the earliest cases--including the sharp practices of Alex's
store-keeping brother-in-law--emerged 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's audit director discovered something odd about the
claims being made by the supervisor of the company's healthcare
department. "The first two digits of the healthcare payments were
checked for conformity to Benford's law, and this revealed a spike in
numbers beginning with the digits '65'," says Nigrini. "An audit
showed 13 fraudulent cheques for between $6500 and $6599...related to
fraudulent heart surgery claims processed by the supervisor, with the
cheque ending up in her hands."

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

Not surprisingly, big businesses and central governments are now also
starting to take Benford's law seriously. "Digital analysis is being
used by listed companies, large private companies, professional firms
and government agencies in the US and Europe--and by one of the
world's biggest audit firms," 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's 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't do them much good--apart from warning
them off: "The problem for fraudsters is that they have no idea what
the whole picture looks like until all the data are in," says
Nigrini. "Frauds usually involve just a part of a data set, but the
fraudsters don't know how that set will be analysed: by quarter, say,
or department, or by region. Ensuring the fraud always complies with
Benford's Law is going to be tough--and most fraudsters aren't rocket

In any case, says Nigrini, there is more to Benford's 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's law.

Ted Hill at Georgia Tech thinks that the ubiquity of Benford's law
could also prove useful to those such as Treasury forecasters and
demographers who need a simple "reality check" for their mathematical
models. "Nigrini showed recently that the populations of the
3000-plus counties in the US are very close to Benford's law," says
Hill. "That suggests it could be a test for models which predict
future populations--if the figures predicted are not close to
Benford, then rethink the model."

Both Nigrini and Hill stress that Benford's law is not a panacea for
fraud-busters or the world's data-crunching ills. Deviations from the
law's 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: "Every mathematical theorem or statistical test can be
misused--that does not worry me," says Hill.

But they share a sense that there are some really clever uses of
Benford's law still waiting to be dreamt up. Says Hill: "For me the
law is a prime example of a mathematical idea which is a surprise to
everyone--even the experts."

Robert Matthews is Science Correspondent for The Sunday Telegraph
Here, there and everywhere

NATURE'S PREFERENCES for certain numbers and sequences has long
fascinated mathematicians. The so-called Golden Mean-- roughly equal
to 1.62 and supposedly giving the most aesthetically pleasing
dimensions for rectangles--has been found lurking in all kinds of
places, from seashells to knots, while the Fibonacci sequence--1, 1,
2, 3, 5, 8 and so on, every figure being the sum of its two
predecessors--crops up everywhere in nature, from the arrangement of
leaves on plants to the pattern on pineapple skins. Benford's 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 x log2 (30
per cent) of such numbers will begin with "1"; 100 x log1.5 (17.6 per
cent) with "2"; down to 100 x log1.11 (4.6 per cent) with "9". But
the mathematics of Benford's law goes further, predicting the
proportion of digits in the rest of the numbers as well. For example,
the law predicts that "0" is the most likely second digit--accounting
for around 12 per cent of all second digits--while 9 is the least
likely, at 8.5 per cent. Benford's law thus suggests that the most
common non-random numbers are those starting with "10...", which
should be almost 10 times more abundant than the least likely, which
will be those starting "99...". As one might expect, Benford's 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's 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's law.

Further reading:

* Digital Analysis Tests and Statistics, written and published
by Mark Nigrini, is available from mark_nigrini@msn.com
* Eric Weisstein's Treasure Troves of Science - Benford's Law page

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