The Benford detector.

Pick a county's population, a company's revenue, the length of a river, the number on your last electricity bill. Intuition says the first digit should be anything from 1 to 9 with equal odds — about 11% each. Intuition is wrong. Across a startling range of real-world data, the leading digit is a 1 roughly 30% of the time and a 9 less than 5% of the time, following a precise curve: the probability of leading digit d is log₁₀(1 + 1/d).

It was first noticed by the astronomer Simon Newcomb in 1881 — the early pages of logarithm tables, the ones beginning with 1, were grubbier and more worn than the later pages — and rediscovered and popularised by the physicist Frank Benford in 1938, who checked it against 20,000-odd numbers from river areas to baseball statistics to the atomic weights of elements. Hence Benford's Law, occasionally the Newcomb–Benford law, and the name no one uses, the first-digit law.

The cleanest way to see why is to stop thinking about a number line and start thinking about a log ruler. On a logarithmic scale, the distance from 1 to 2 is the same as from 2 to 4 or 4 to 8 — each is one doubling. Crucially, the stretch occupied by numbers beginning with 1 (from 1 up to 2) is far longer than the stretch beginning with 9 (from 9 up to 10).

This also explains the law's deepest property: it is scale-invariant. Convert every figure from dollars to euros, miles to kilometres, or square feet to hectares — multiply the whole dataset by any constant — and the leading-digit distribution is unchanged. A regularity that survives changing the units can't be an artefact of the units, which is a strong hint it is real. It is also why the powers of 2 and the Fibonacci numbers in the tool above follow it almost perfectly: multiplicative sequences slide along the log ruler at a steady rate, visiting each band in exact proportion to its width.

Because honest, naturally-arising figures tend to obey the law and many invented figures don't, Benford's Law became a staple of forensic accounting. People fabricating numbers reach for digits too evenly, round too neatly, cluster below psychological thresholds, and repeat themselves — all of which bend the first-digit curve. The accounting scholar Mark Nigrini turned this into a practical audit technique, and Benford tests now sit inside tax-authority software, expense-fraud screens and corporate audit tools worldwide. Suspicious patterns in macroeconomic data — including the figures Greece reported before its debt crisis — have been flagged the same way.

The key word, though, is smoke alarm. A failed Benford test is a reason to look closer, not a verdict. Which brings us to the part of the story this site exists to tell.

Benford's Law applies to a specific kind of data: figures that range freely across several orders of magnitude. Vast amounts of perfectly honest data don't qualify, and quietly fail the test for reasons that have nothing to do with fraud. Adult heights in centimetres cluster between 150 and 200, so nearly every leading digit is a 1. Assigned numbers — phone numbers, postcodes, invoice IDs — follow whatever rule assigned them. Anything bounded, rounded, or capped breaks the spell.

This is exactly where Benford's Law gets misused, and the most public example arrived in November 2020, when viral posts “proved” US election fraud by showing that certain candidates' precinct-level vote counts didn't follow Benford's Law. The argument was statistically empty. Precinct vote totals occupy a narrow, bounded range set by how precincts are sized — precisely the conditions under which honest data fails the first-digit test. Election-forensics researchers had warned for years that Benford's Law is unreliable for vote counts, and the 2020 claims were a textbook case of running a valid tool far outside its valid range and reading the failure as a smoking gun.

That is the unspurious lesson in miniature. Benford's Law is real, elegant, and genuinely useful. It becomes an illusion the instant its preconditions are dropped: applied to bounded data, tiny samples, or assigned numbers, it manufactures false suspicion as readily as it once caught real fraud. Use it as a screen that raises questions for a human to investigate — never as a proof that closes them.