Take any long list of real-world numbers: river lengths, share prices, populations, the figures in a company's accounts. Now look only at the first digit of each. You might expect each digit from 1 to 9 to lead about one time in nine. It doesn't. The digit 1 leads about 30% of the time, and the digit 9 only about 5%.
In 1881 the astronomer Simon Newcomb noticed that the early pages of books of logarithm tables were more worn than the later ones. People were looking up numbers starting with 1 far more often than numbers starting with 9. In 1938 the physicist Frank Benford checked the idea on more than 20,000 numbers from about 20 different sources, from river areas to street addresses, and found the same lopsided pattern. It now carries his name.
The rule is simple: the chance that the first digit is d equals log10(1 + 1/d). That gives 30.1% for a 1, 17.6% for a 2, and so on down to 4.6% for a 9.
Pick a list below. The bars show how often each digit leads; the lime markers are Benford's prediction.
Powers of 2, Fibonacci numbers, factorials and long chains of multiplied random factors all land close to Benford's curve. Numbers drawn evenly from 10 to 99 give each digit about the same share, and simulated adult heights (nearly all between 100 and 199 cm) start with 1 almost every time.
Now drag the multiplier. Convert the numbers from dollars to euros, or metres to feet, and the spread-out lists keep their shape. The narrow ones fall apart: stretch the heights by 1.3 and the leading 1s turn into 2s.
The trick is to stop thinking of numbers on an ordinary ruler and picture them on a logarithmic one, where each step of ×10 takes the same length. On that ruler the stretch of numbers from 1 to 2 is wide, and the stretch from 9 to 10 is thin.
If a quantity grows by a steady percentage, say 7% a year, it moves along this ruler at a steady speed. It therefore spends the same amount of time in each stretch of the ruler, and the stretch for the digit 1 is the longest, so most of the time the number begins with a 1. A balance growing from 1,000 to 2,000 sits on a leading 1 the whole way, while the climb from 8,000 to 9,000 takes much less time at the same growth rate.
The same logic explains why the rule survives a change of units. Multiplying by any constant only slides everything along the ruler, and a list that fills the ruler evenly looks the same after sliding. Benford's law is the only first-digit pattern with that scale-free property.
People inventing figures tend to spread their first digits roughly evenly, which is exactly what real data doesn't do. Forensic accountants and tax auditors therefore compare the first digits of a ledger with Benford's curve. A mismatch isn't proof of fraud, but it is a cheap signal that a closer look is worth the time.
The method has limits. It needs numbers that cover several orders of magnitude, so heights, ages and exam scores don't qualify. Numbers assigned by a rule, such as phone numbers, postcodes or prices fixed at 9.99, don't qualify either. Used on the right kind of data, a lopsided first digit is a good reminder that nature, finance and mathematics tend to grow by multiplying, not by adding.