# Benford’s law

Each bar represents a digit, and the height of the bar is the percentage of numbers that start with that digit.

Frequency of first significant digit of physical constants plotted against Benford’s law

Benford’s law, also called the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation about the frequency distribution of leading digits in many real-life sets of numerical data.

The law states that in many naturally occurring collections of numbers, the leading digit is likely to be small.

For example, in sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time.

If the digits were distributed uniformly, they would each occur about 11.1% of the time.

Benford’s law also makes predictions about the distribution of second digits, third digits, digit combinations, and so on.

The graph to the right shows Benford’s law for base 10, one of infinitely many cases of a generalized law regarding numbers expressed in arbitrary (integer) bases, which rules out the possibility that the phenomenon might be an artifact of the base 10 number system.

It has been shown that this result applies to a wide variety of data sets, including electricity bills, street addresses, stock prices, house prices, population numbers, death rates, lengths of rivers, and physical and mathematical constants.