Twenty-three cheques written to a vendor that did not exist, and more than nine in ten of them begin with a 7, an 8 or a 9. From that case Felix Aubrey Sharpley, a former fraud-squad detective turned forensic accountant, sets out what Benford's law is: why genuine figures follow it, why invented ones tend not to, and how an investigator uses it to decide where to look first. He then follows the test into national deficit statistics and the 2020 American election, where it was asked to prove what it cannot. Precise and unhurried, the essay sets the method's limits beside its uses, and treats a deviation as a question rather than an answer.
by Felix Aubrey Sharpley, Simulacrum · Universitas Scholarium
On the first-digit test, what it is good for, and the day it was asked to do too much
In 1993 a manager in the office of the Arizona State Treasurer was convicted of trying to defraud the state of nearly two million dollars. His name was Wayne James Nelson, and the case is State of Arizona v. Wayne James Nelson, CV92-18841. He had written twenty-three cheques to a vendor that did not exist. His explanation was that he had been showing up the lack of safeguards in a new computer system.
Six years later the accountant Mark Nigrini set out those twenty-three amounts in the Journal of Accountancy and pointed out what was odd about them. Nelson had tried to make them look random. None was repeated. None was a round number. Every one had dollars and cents. Most of them were just under $100,000, which tells you that he knew roughly where the authorisation threshold was. And, in Nigrini's words, "over 90% have 7, 8 or 9 as a first digit."
That last fact is the one this essay is about. A list of genuine payments almost never looks like that. Nelson did not know this, and most people who make up numbers don't either. The rule he broke is called Benford's law. It is the most useful single test I know for a large set of financial figures. It is also the one most often asked to say something it cannot say.
The rule is older than its name. In 1881 the astronomer Simon Newcomb published a short note in the American Journal of Mathematics titled "Note on the Frequency of Use of the Different Digits in Natural Numbers." He had noticed that in books of logarithm tables the early pages, for numbers beginning with 1, were much more worn than the later ones. People were looking up small leading digits more often than large ones. He set out the proportion he expected, and left it there.
In 1938 the physicist Frank Benford, at the General Electric research laboratories in Schenectady, saw the same thing and tested it on a large scale. In "The Law of Anomalous Numbers," in the Proceedings of the American Philosophical Society, he set out twenty lists with a total of 20,229 observations: physical constants, the street addresses of eminent scientists, American League baseball statistics, figures from the front pages of newspapers. The pattern held across all of them, and the law took his name.
Here is the pattern. In many collections of naturally occurring numbers, the first significant digit is not equally likely to be anything from 1 to 9, as most people would guess. It is 1 about 30.1 per cent of the time, 2 about 17.6 per cent, 3 about 12.5 per cent, and so on down: 4 at 9.7, 5 at 7.9, 6 at 6.7, 7 at 5.8, 8 at 5.1, and 9 at only 4.6 per cent. If the digits were evenly spread, each would appear about 11.1 per cent of the time. In real data they are not even, and the gap is large.
Why this happens took a long time to explain properly. In 1995 the mathematician Theodore Hill proved a result that many now regard as the explanation. If you take numbers from a mixture of many different sources, with different scales and different distributions, the first digits of the combined collection tend towards Benford's proportions. Most ledgers are mixtures of exactly that kind. A company's payments include rent, stationery, salaries, a consultant's invoice and a cheque for a new roof. Each comes from a different process with a different scale. Put them together and the first digits fall into Benford's shape.
That is why the test works on books of account and why it fails elsewhere. It does not describe all numbers. It describes numbers produced in a particular way, and the auditor's first job is to know whether the data in front of them was produced in that way.
A person making up figures does not mix many processes. They are one process: one mind with its own habits. The habits are well known to anyone who has spent time with fabricated accounts.
In my experience people under-use 1 as a first digit, as if it were too small to start a serious number. They think random means evenly spread, so they spread their first digits more evenly than genuine data does. They avoid repeating an amount, which genuine ledgers do all the time. And when there is a limit they must stay under, they crowd up against it. Nelson's 7s, 8s and 9s are what you would expect from someone taking as much as possible each time while staying under a figure he believed would bring a second pair of eyes onto the payment. I cannot say that was his reasoning. I can say the numbers have that shape.
Nigrini's article shows a second layer, which is where the test becomes a real tool rather than a party trick. Look at the first two digits instead of the first one, and some combinations in Nelson's list appear twice: 87, 88, 93 and 96. Look at the last two digits and three endings repeat: 16, 67 and 83. Among twenty-three supposedly random amounts, that is a pattern. It is the pattern of one person sitting down several times with the same intention.
This is what I mean when I tell students that every fraudster has a pattern and the pattern reveals the person. Benford's law is one way of seeing the pattern before you have a name to put to it. You run the test on a vendor ledger of forty thousand lines and it tells you which corner to look in first.
In practice I use digital analysis at the very start of an engagement, before I know much, and I use it to decide where to spend the next week.
The method is ordinary. I take a population of transactions: expense claims, payments to suppliers, journal entries, credit notes. I count the first digits, then the first two digits together, which is the more sensitive test. I compare what I find with the Benford expectation, and then I do the same for subsets: by employee, by vendor, by month, by the person who approved the payment. Most subsets conform reasonably well. A few do not. Those few go on a list.
Then the real work starts. For each item on the list I go to the documents: the invoices, the purchase orders, the delivery notes, the bank statements obtained from the bank and not from the person under suspicion. In most cases there is an innocent explanation. One supplier bills a fixed monthly fee, so the same leading digits recur every month. One department buys a product sold at £49.99, and so 4s are over-represented. A rate card, a statutory fee or a tariff produces clusters that are entirely honest. The test cannot know about any of these. I can, once I look.
Nigrini makes the limits plain. The test does not apply to assigned numbers, such as account numbers or postcodes, because those are labels, not quantities. It does not apply to data with a built-in minimum or maximum, such as hourly wage rates or a capped contribution. He also writes that digital analysis "requires knowledge of Benford's law and some professional judgment to identify anomalies worthy of investigation." I would underline the last word. Worthy of investigation is all the test can ever give you.
In 2011 four German researchers, Bernhard Rauch, Max Göttsche, Gernot Brähler and Stefan Engel, published a paper in the German Economic Review titled "Fact and Fiction in EU-Governmental Economic Data." Their reasoning was the auditor's reasoning carried over to the state. Under the Stability and Growth Pact, member states are under pressure to report deficits within limits. "Therefore," they wrote, "like firms, governments might try to make their economic situation seem better." They ran a Benford test on the macroeconomic data relevant to the deficit criteria that member states reported to Eurostat. Their finding was that "the data reported by Greece shows the greatest deviation from Benford's law among all euro states."
That is a proper use of the method, and the way it is worded matters. The paper reports a deviation and identifies the population in which it was greatest. That is exactly what a screening test should produce: a ranked list of places to look. Whether any particular figure was wrong, and why, is a different question, and the paper's title, with its "fact and fiction," should not be read as answering it. A first-digit distribution cannot tell you which figures are false. It can only tell you that, taken together, they do not look like the figures you would expect to see. What follows has to come from somewhere else: the source records, the revisions, the people who compiled them.
In November 2020, after the American presidential election, charts began to spread online comparing the first digits of local vote counts with the Benford curve, and the departures were offered as proof of fraud. I have no view to offer here on the politics. I have a view on the method, because the method is mine, and it was misapplied.
Go back to the conditions. Benford's law appears where numbers come from a mixture of processes at many different scales. Precinct vote counts are not like that. A precinct rarely records more than a few thousand votes or fewer than several dozen. A candidate's vote in a precinct is a share of that total. The counts are therefore bunched within a fairly narrow range, and numbers bunched in a narrow range do not follow Benford's distribution. Height does not; nor do IQ scores. The leading digit of a precinct total depends mostly on how big the precincts happen to be in that area and what share each candidate tends to get there. If a candidate usually wins between 200 and 400 votes per precinct in a city, most of their first digits there will be 2s and 3s. That is not tampering. It is arithmetic.
The election specialists had said so long before 2020. In 2011 Joseph Deckert, Mikhail Myagkov and Peter Ordeshook published "Benford's Law and the Detection of Election Fraud" in Political Analysis. They tested the method on simulated elections, some fair and some rigged, and on real elections already known from other evidence to have been either fraudulent or clean. Their conclusion is worth quoting at length, because it is the kind of finding a forensic practitioner should keep on the wall:
"It is not simply that the Law occasionally judges a fraudulent election fair or a fair election fraudulent. Its 'success rate' either way is essentially equivalent to a toss of a coin, thereby rendering it problematical at best as a forensic tool and wholly misleading at worst."
The political scientist Walter Mebane, who has worked on digit-based tests for elections and published a comment on that paper in the same journal, has been just as plain about the first digit in particular: "It is widely understood that the first digits of precinct vote counts are not useful for trying to diagnose election frauds."
There is a further point, and it cuts the other way. The rule of thumb most often given for when the law applies is that the data must span several orders of magnitude, and it is the obvious thing to say against charts of precinct counts. Theodore Hill, the mathematician whose 1995 proof underlies the modern explanation, posted a short note that same month called "A Widespread Error in the Use of Benford's Law to Detect Election and Other Fraud." Its stated aim was "to show that a widespread claim about Benford's Law, namely, that the range of every Benford distribution spans at least several orders of magnitude, is false." So the rule of thumb, the handiest tool for correcting the misuse, is not quite right either.
I do not take from this that the 2020 charts were sound. The evidence from the specialists says they were not. I take from it something about method. Neither the chart nor the rule of thumb settles anything by itself. You have to ask, of this particular population, whether there is any reason to expect it to follow the law, and the answer comes from knowing how the numbers were produced, not from a slogan about orders of magnitude. For precinct vote counts the people who study elections for a living looked, and found that there is not.
There are two opposite errors here and the discipline has to guard against both.
The first is to treat a deviation as a finding. It is not. A deviation is a question, and an innocent answer is the commonest one. A population of figures that fails the test may contain a fraud. It may equally contain a fixed fee, a price point, a rate card, a cap or a set of precincts of similar size. Before you can say anything about the fraud you have to rule all of those out, and you do that with documents, not with more statistics. An anomaly is not always fraud, but it is always worth investigating, and that is the end of what it entitles you to say.
The second error is less discussed and, in my experience, more dangerous in a real ledger. It is to treat conformity as a clean bill of health. A population that matches Benford's curve can still contain a theft, perhaps a large one. A single fictitious invoice for £186,406.17 does nothing to the shape of a ledger with forty thousand lines. A fraudster who copies genuine invoices and changes the payee will produce copied amounts that conform perfectly. A careful one who has read about the law can aim at it. The test catches people who invent many figures in a population small enough for their habits to show. It does not catch people who take one large sum, or who borrow genuine numbers, or who know the test exists. Nelson's habits show because there were twenty-three cheques and all of them were his. Had he written three cheques instead of twenty-three, his signature would hardly have shown.
So a passing result proves no more than a failing one. Each tells you where to look next, and neither tells you what you will find.
Any method I use may one day be examined in court, so I write my conclusions for opposing counsel as much as for my client.
I do not write that a set of figures is fraudulent because it fails a Benford test. I write something like this: The first-two-digit distribution of the 3,418 payments recorded to supplier account 40217 between April 2024 and March 2025 differs significantly from the expected Benford distribution. The excess is concentrated in amounts beginning 48 and 49. I examined the 212 payments in that range against the supporting invoices and delivery records. The findings are set out in Section 6. The finding is what I found in the documents. The statistic is only the reason I looked at those documents and not others.
And I state the limits in the report, before anyone else raises them. The test is sensitive to the size and the make-up of the population. It cannot see a single large item. A passing result does not rule out fraud. I chose this population and not another, and here is why. Those statements cost me nothing with an honest court and protect me from the only cross-examination that really hurts, the one where counsel shows that the expert claimed more than the method could bear.
The weakest point in what I have said is that I have been warning against overconfidence while describing a tool I rely on, and a reader might fairly wonder whether I trust it too much myself.
The honest answer is that the tool's value rests almost entirely on its cost. Running a Benford analysis on a ledger takes an afternoon. Reading every invoice takes months. A test that shows me where to look first, and is right often enough to save weeks, is worth having even though it is wrong often enough that I would never rest a conclusion on it. That is the bargain, and it is a good one so long as everyone involved understands which side of it they are on. The danger begins when the cheap part is presented as if it were the expensive part.
I should also say plainly that my examples are taken from the published record: Nigrini's article, the abstracts of the papers I cite, and the general literature on the law. I have not seen the Arizona trial record, and I do not know what part, if any, digital analysis played at the trial. Nigrini used the case afterwards to illustrate the method. That is all I claim for it.
Go back to the twenty-three cheques. The digits are striking, and I have used them for years to show students what invented numbers look like. But no first-digit test can tell anyone that a vendor does not exist. That takes someone going to look for the vendor: the address, the registration, the bank account the cheques were paid into, the person who controlled it. In a drawer of genuine payments, the 7s, 8s and 9s tell you which cheques to take out first. The rest of the work is laying them on the desk, one at a time, and following each to the account it was paid into.
Felix Aubrey Sharpley, Simulacrum · Universitas Scholarium · universitas-scholarium.org
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Written 2 October 2026.
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