Benford’s Law in Financial Statements: Can Digit Patterns Spot Doctored Numbers?

Benford’s law in financial statements sounds like something only forensic auditors should care about. It is not. The idea is simple enough to test on a spreadsheet: in many sets of real-world numbers, the first digit is far more often a 1 than a 9, and numbers that somebody has made up tend to break that pattern. If you read company results and wonder whether the figures in front of you were born naturally or were massaged into shape, this is one of the oldest and cheapest checks there is.

We will be honest about it up front. A Benford test does not prove fraud. It tells you where to look. That distinction matters a lot, and we will come back to it.

What Benford’s law says

Pick a large set of numbers that describe sizes of things, such as invoice amounts, town populations or company market values, and look only at the first digit of each one. Your instinct says each digit from 1 to 9 should show up about one time in nine, or roughly 11.1%. That instinct is wrong for a surprising amount of data.

According to the Wikipedia entry on Benford’s law, a set satisfies the law if the leading digit d turns up with probability log₁₀(1 + 1/d). Run that formula and you get this table:

So a 1 leads about 30% of the time and a 9 less than 5% of the time. Add the first three rows and you get 60.2%, which is the figure Mark Nigrini gives in his 1999 Journal of Accountancy article “I’ve Got Your Number” for the chance that the first digit is a 1, 2 or 3.

The law also makes predictions for the second digit, the third digit and combinations such as the first two digits together. Auditors use those finer tests a great deal, as we will see.

Where it came from

The story starts with worn-out books. In 1881 the astronomer Simon Newcomb noticed that the early pages of logarithm tables, the ones covering numbers starting with 1, were much more worn than the later pages. He published a note proposing the same logarithmic rule.

The physicist Frank Benford noticed the same thing and, in a 1938 paper titled “The Law of Anomalous Numbers”, tested it on data from 20 different domains: river surface areas, populations, physical constants, molecular weights, numbers from an issue of Reader’s Digest and more, 20,229 observations in all. The law ended up carrying his name rather than Newcomb’s. Both details come from the Wikipedia history section linked above, and Nigrini’s article describes the same 20 lists and 20,229 observations.

Why real numbers lean towards 1

The cleanest explanation is growth. Nigrini uses a mutual fund example, and it translates neatly into rupees because the law does not care about currency.

Suppose a fund has assets of Rs 100 crore and grows at 10% a year. The first digit stays 1 until assets reach Rs 200 crore. That needs a 100% increase, which at 10% compounded takes about 7.3 years. Now start at Rs 500 crore. Getting to Rs 600 crore needs only a 20% rise, about 1.9 years. From Rs 900 crore to Rs 1,000 crore is an 11.1% rise, about 1.1 years. Then the first digit is 1 again, and it sits there for another 7.3 years.

You can check the arithmetic yourself: the years needed are ln(2)/ln(1.1), ln(1.2)/ln(1.1) and ln(10/9)/ln(1.1). Nigrini’s article gives the same 7.3, 1.9 and 1.1 years. Anything that grows at a steady or even an erratic rate spends more of its life with a low leading digit.

There is a second reason the law works on financial data across countries. Nigrini points to Roger Pinkham’s 1961 result that Benford’s law is scale invariant: multiply every number in a conforming set by a constant and it still conforms. That is why converting dollars to rupees, or lakh to crore, does not break the pattern.

A Benford test never tells you a number is false. It tells you which pile of numbers deserves a second, slower look.

Using Benford’s law in financial statements and ledgers

Here is the key point for anyone applying it to company accounts. The headline numbers in a quarterly result, revenue, EBITDA, net profit, are a handful of figures. You cannot run a meaningful digit test on ten numbers. Benford analysis earns its keep on large transaction files: accounts payable, expense claims, journal entries, sales invoices, write-offs. That is why it is mainly a tool for auditors and investigators who have access to the ledger, not for an investor reading a results press release.

Nigrini’s article lists the typical uses: accounts payable data, estimates in the general ledger, duplicate payments, system conversions, customer refunds and processing inefficiencies. He also explains the human side. People who invent numbers do not naturally favour low digits, so made-up amounts tend to drift away from the expected pattern.

The Arizona cheques

The best-known example in Nigrini’s article is a 1993 case, State of Arizona v. Wayne James Nelson. A manager in the office of the Arizona State Treasurer was found guilty of trying to defraud the state of nearly $2 million by diverting funds to a bogus vendor through 23 cheques.

The amounts looked random on purpose. None was repeated, none was a round number and all included cents. But the digit pattern was almost the opposite of Benford’s: more than 90% of the cheques began with 7, 8 or 9. Most amounts sat just below $100,000, which Nigrini suggests may have been a threshold for extra scrutiny. Across the 160 digits used in the 23 amounts, 7, 8 and 9 were the most common. A person trying to look random had leaned on high digits without realising it.

Spikes just under a limit

The first-two-digit test is where a lot of real findings come from. Nigrini gives several:

• One company’s expense data showed a spike at the digits 24. The cause was claims of $24.50, just under a $25 limit above which a voucher was needed.
• In a bank audit, credit card balances written off as uncollectible showed a large spike at 49. Most of it came from amounts between $4,800 and $4,999, and one officer was responsible for the bulk of them. The internal write-off limit was $5,000. Friends of the officer would run up balances just below that and he would write them off.
• Managers with a purchase approval limit sometimes bunch invoices just below it, which shows up as spikes in the digits just under the limit.

Not every spike is crime. In one accounts payable file, the most frequent amounts of $25, $30 and $10 turned out to be courier charges. At another company, the test pointed to roughly 12,000 invoices a year from the same vendor for employee business cards, a processing cost that monthly billing could cut. The digits found waste, not theft.

A worked example you can repeat in Excel

Let us make this concrete with a hypothetical file. These figures are made up for illustration only; they do not belong to any company.

Say an internal auditor pulls 1,000 vendor payments from a mid-sized company’s books and counts the first digit of each amount. Expected counts are just 1,000 multiplied by the Benford share for each digit.

The last column is the actual share of payments starting with that digit minus the Benford share, in percentage points. Two things jump out. Digit 4 is 4.5 points too high, and digit 1 is 3.9 points too low. Everything else is close.

A common summary number is the mean absolute deviation: take the absolute value of each digit’s gap, add them up and divide by nine. Here it works out to about 1.45 percentage points. For comparison, Nigrini reports a mean absolute deviation of 0.7% for the first digits of population counts in 3,141 US counties from the 1990 census, data that followed the law closely and which, he says, auditors would usually treat as an immaterial difference. Our hypothetical file is about twice as far off.

So what does the auditor do? Not accuse anyone. They drill into the payments starting with 4, then run the first-two-digit test on them. Suppose most of the excess sits at 48 and 49, meaning amounts between Rs 48,000 and Rs 49,999. If the company’s rule is that payments of Rs 50,000 and above need a second approver, the pattern now has a plausible story: someone may be splitting bills or pricing just under the limit. That is a lead for a document check, nothing more.

To run this yourself, put the amounts in column A, use =VALUE(LEFT(TEXT(INT(A2),”0″),1)) to pull the first digit for amounts of 1 or more, count each digit with COUNTIF, and compare with =LOG10(1+1/d). It takes ten minutes.

Tests auditors layer on top

Nigrini’s article describes a few checks that sit alongside the digit tests and tend to find plain errors rather than bias:

• Relative size factor. For each vendor, divide the largest payment by the second largest. A very high ratio flags an outlier. In one case, $452.47 had been keyed as $45,247 because of a misplaced decimal point, and the high ratio caught it.
• Same, same, same. Identical entries, which often mean duplicate payments.
• Same, same, different. Same invoice number and same amount but a different vendor number, which can mean the wrong vendor was paid first and the right one later.

On the statistics, the Wikipedia article notes that the chi-squared test is often used to check conformity but has low statistical power with small samples, and that tests such as Kolmogorov-Smirnov and Kuiper are more powerful there. The practical lesson is the same either way: the smaller the file, the less a digit test can tell you.

The data that suits Benford best is messy, wide-ranging and big. Neat, capped or assigned numbers will fail the test without anyone having done anything wrong.

When Benford’s law does not apply

This is where most amateur uses go wrong. Nigrini says data most likely to follow the law describes the sizes of similar things, such as market values of companies, and has no built-in maximum or minimum. Assigned numbers such as account numbers, PIN codes or invoice serials will not conform. Neither will figures with a hard cap or a fixed rate.

The Wikipedia entry adds that the law works best when data spans several orders of magnitude, and that a plain uniform distribution and the normal distribution do not obey it. Think of salaries in a single pay grade, prices set at Rs 499 or Rs 999 for marketing reasons, or a fixed fee charged to every customer. Each will fail a Benford test, and none of that is fraud.

There is also a well-known caution from election studies. Wikipedia notes that there is no scientific consensus that Benford’s law reliably detects election fraud, and cites a 2011 study arguing it can mislead as a fraud indicator. Accounting data is different from vote counts, but the warning travels: a mismatch is a question, not a verdict.

What this means for an investor reading results

You will rarely have a company’s ledger. What you can do is use the idea behind Benford, that real numbers have a texture and invented ones often do not, as one more reason to look harder at a company whose numbers seem too smooth or too neatly clustered.

In practice, the investor’s toolkit for spotting manipulated results is still mostly ratio-based. Our guide to the Beneish M-Score covers an eight-variable model built for exactly this question, and our piece on revenue recognition under Ind AS 115 explains where aggressive booking of sales usually hides. For the wider routine of reading a quarterly report, start with how to analyse quarterly results of a company.

Benford fits in that kit as a screen, not a judge. If you ever get a large transaction-level dataset, say from an annual report annexure, a government disclosure or your own business, it is worth ten minutes of Excel. If the digits misbehave, ask why. Usually the answer is a limit, a fixed price or a quirk of the data. Occasionally it is something that needs a much closer look.

Sources

• Mark J. Nigrini, “I’ve Got Your Number”, Journal of Accountancy, May 1999
• Wikipedia: Benford’s law

Disclaimer: This article is for information and education only and is not investment advice. Please consult a SEBI-registered adviser before making any investment decision.