Benford's Law | mysimulator.uk

A law about the first digit

Benford's Law says that in many naturally occurring collections of numbers, the leading digit is not uniformly distributed among 1 through 9 — it follows a sharply skewed logarithmic curve. The digit 1 leads about 30.1% of the time, 2 about 17.6%, all the way down to 9, which leads only about 4.6% of the time. The formula, first noted by astronomer Simon Newcomb in 1881 and rediscovered by physicist Frank Benford in 1938 after he compiled over 20,000 numbers from rivers, populations and physical constants, is:

P(first digit = d) = log10(1 + 1/d),   d = 1, 2, ..., 9

P(1) ≈ 0.301   P(2) ≈ 0.176   P(3) ≈ 0.125
P(4) ≈ 0.097   P(5) ≈ 0.079   P(6) ≈ 0.067
P(7) ≈ 0.058   P(8) ≈ 0.051   P(9) ≈ 0.046

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The intuition is scale invariance. If a dataset spans several orders of magnitude — country populations from a few thousand to over a billion, or the Fibonacci sequence, which grows exponentially — then on a logarithmic axis the numbers are roughly uniformly spread. A number's leading digit is determined by where its logarithm falls within its integer part, and the interval of log-space that starts with digit 1 (from log10(1) to log10(2)) is wider than the interval that starts with digit 9 (from log10(9) to log10(10)). Multiplying by any constant shifts the log-uniform distribution but does not change its shape, which is why the law holds for data in dollars, euros or any other unit — a property called scale invariance.

Which datasets obey it, and which don't

Benford's Law tends to hold for data generated by multiplicative growth processes spanning several orders of magnitude: populations, river lengths, stock prices, physical constants, powers of two, and the Fibonacci sequence (whose ratio of successive terms converges to the golden ratio, guaranteeing a fixed, non-integer increment in log-space each step). It reliably fails for data that is capped in range, assigned rather than measured, or rounded to a narrow band — telephone area codes, human heights in centimetres, lottery numbers, or any list confined to two or three orders of magnitude.

Catching fraud with a chi-squared test

Benford's Law became famous outside mathematics as a forensic accounting tool. Fabricated numbers — invented expense reports, falsified tax filings, doctored election tallies — are usually generated by people trying to look random, and human-picked digits cluster toward the middle of the range (5, 6, 7) far more than genuine multiplicative data does, which is heavily front-loaded onto 1 and 2. Auditors compare the observed first-digit frequencies in a real ledger against Benford's predicted frequencies using a chi-squared goodness-of-fit test:

chi_sq = sum( (observed_d - expected_d)^2 / expected_d )  for d = 1..9

expected_d = N * log10(1 + 1/d)     # N = sample size
large chi_sq, small p-value  → distribution deviates from Benford
                                (worth a closer look, not proof of fraud)

A large deviation is a red flag that prompts investigation, not a verdict — plenty of legitimate datasets fail the test simply because they don't meet the scale-invariance precondition, and the test has genuinely been used both to uncover accounting fraud (it played a role in some well-known corporate cases) and to argue about anomalies in election returns, where its applicability is far more contested because vote counts are bounded by the number of registered voters rather than open-ended.

Second-digit and higher-order extensions

The same logic extends to the second digit, the digit pairs, and even the leading digits of ratios between rows in a dataset, each with its own predicted distribution derived from the same log10 formula (the second-digit distribution is flatter, converging toward uniform as you look further from the leading digit). Investigators layer several of these tests together, since a fabricator who successfully mimics the first-digit distribution rarely also gets the second-digit and digit-pair distributions right by chance.

Frequently asked questions

Why does Benford's Law apply to Fibonacci numbers but not to random dice rolls?

The Fibonacci sequence grows multiplicatively — each term is roughly phi times the previous one — so its logarithms increase by a fixed step and spread uniformly across many orders of magnitude, which is exactly the condition under which the leading-digit formula log10(1+1/d) holds. Dice rolls are bounded to 1-6 with no multiplicative structure, so they stay uniformly distributed instead.

Does failing a Benford's Law test prove a dataset was faked?

No. It is a screening tool, not proof. Many legitimate datasets fail simply because they don't span enough orders of magnitude or aren't generated by a multiplicative process — human heights and telephone numbers are honest data that never obey Benford's Law. A deviation flags a dataset for closer manual review.

Why is the digit 1 so much more common than 9 as a leading digit?

On a logarithmic scale, the interval covering numbers that start with 1 (from 1 to just under 2) is proportionally the widest span before the next power of ten, while the interval for numbers starting with 9 (9 to 10) is the narrowest. Since scale-invariant data is roughly log-uniform, more of it falls into the wide '1' interval than the narrow '9' interval.

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Benford's Law: Why the Digit 1 Beats the Digit 9 Six to One