Pick any number from a newspaper, tax return, or list of river lengths. You'd think the first digit is random. In reality, nearly a third of them start with 1, and only 4.6% start with 9.
This counterintuitive phenomenon is known as Benford's Law, or the law of anomalous numbers. It dictates that in many naturally occurring collections of numbers, the leading digit is likely to be small. The digit 1 appears as the first digit about 30.1 percent of the time, 2 appears 17.6 percent of the time, and the probabilities steadily decrease until the digit 9, which appears less than 5 percent of the time.
The reason this happens comes down to how things grow. Many systems in the natural and human worlds grow exponentially or multiplicatively, rather than by adding a fixed amount. For example, if a city has a population of 100,000 and grows at a rate of 5 percent a year, it takes a long time to reach 200,000. During all those years, the population figure starts with a 1. Once it finally reaches 200,000, growing at that same 5 percent rate means it will reach 300,000 much faster. By the time the population reaches 900,000, a 5 percent increase adds 45,000 people in a single year, blasting through the 900,000s and rolling over to 1,000,000 very quickly. The number spends far more time starting with a 1 than it does starting with an 8 or a 9.
Benford's Law applies to a staggering variety of datasets, including:
- The populations of cities and counties
- Stock market prices and trading volumes
- The lengths of rivers and heights of mountains
- Accounting figures and corporate expenses
Because this pattern is so pervasive, forensic accountants and auditors actively use Benford's Law to catch fraud. When people try to invent fake numbers for tax returns or expense reports, they usually try to make the data look random by evenly distributing the first digits. A fraudster might assume that starting fake expenses with a 7, 8, or 9 is just as unsuspicious as starting them with a 1 or 2. An auditor analyzing those figures will immediately notice the lack of 1s and the overabundance of higher digits, flagging the documents as artificially fabricated. The very attempt to make the numbers look natural is exactly what exposes them as fake.