Benford’s Law: Why So Many Real Numbers Start With 1

A pattern hiding in plain sight

If you picked a number at random from a newspaper, a company ledger, or a list of city populations, you might expect its first digit to be anything from 1 to 9 with equal odds — roughly 11% each. In many collections of real numbers, that is not what happens. The digit 1 leads about 30% of the time, while 9 leads less than 5% of the time.

The astronomer Simon Newcomb noticed this in 1881 when he saw that the early pages of logarithm tables — the ones for numbers beginning with 1 — were far more worn than the later pages. The physicist Frank Benford tested the idea in 1938 on more than 20,000 numbers drawn from sources as varied as river areas, population figures, and physical constants. The pattern held often enough that it now carries his name.

The expected shares

Benford’s Law predicts the share of numbers whose first digit is d with a short formula:

P(d) = log₁₀(1 + 1/d)

Notice that the first three digits together account for about 60% of leading digits, while 7, 8, and 9 together account for only about 15%.

Why it happens

The intuition comes from growth. Imagine a balance that grows by a steady percentage each year. Going from 100 to 200 means doubling, so the number spends a long time with a leading 1. Going from 900 to 1,000 needs only about 11% growth, so the leading 9 disappears quickly. Quantities that grow multiplicatively, or that span several orders of magnitude, therefore spend more of their ‘life’ starting with small digits.

Another way to put it: Benford’s Law is what you get when numbers are spread evenly on a logarithmic scale rather than an ordinary one.

How auditors use it

Researchers such as Mark Nigrini popularized Benford analysis in forensic accounting during the 1990s. The idea is simple: when people invent figures, they tend to spread first digits too evenly or overuse ‘comfortable’ digits. Comparing a set of invoices or expense claims with the expected Benford shares can highlight batches that deserve a closer look.

That word matters: a closer look. A Benford mismatch is a screening signal, not evidence of wrongdoing. Plenty of honest datasets do not follow the law at all.

When Benford’s Law does not apply

• Assigned numbers, such as phone numbers, ZIP codes, or invoice IDs, carry no natural magnitude.
• Data with a narrow range, such as adult heights or exam scores out of 100, cannot span orders of magnitude.
• Capped or minimum values, such as a $25 expense limit or prices set just under a threshold, distort the leading digit.
• Small samples are noisy; a few dozen numbers can drift far from the expected shares by chance.

To try it yourself, count the first digits in a few hundred figures from a source that spans a wide range — monthly bank transactions over several years works well — and convert each count to a percentage of the total.