# The Center of Math Blog

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## Thursday, October 23, 2014

### Throwback Fact of the Week - Benford's Law - 10/23/14

Benford's Law, or more descriptively, the first-digit law, states that in most number lists and real-life collections of data, (such as death rates, baseball statistics, etc.) 1 occurs in the leftmost leading position roughly 30% of the time. It also asserts that larger digits occur less and less frequently, with 9 occurring a mere 4.6% of the time as the first digit.

The significance of this law lies in the fact that it shows that the most common first digit in an arbitrary source of data is not random; if it were random, every digit would be expected to occur first about 11.1% of the time, a 1 out of 9 probability.

The graph below shows probability that each digit occurs as the leading digit. A given set of numbers satisfies Benford's Law if the leading digit d, where d can be the digits 1 through 9, occurs with probability log10(1+1/d).

The law is named after physicist, Dr. Frank Benford, who stated it in 1938. However, it had been previously pointed out by Simon Newcomb in 1881, when he noticed that the pages of logarithms containing the numbers beginning with 1 were much more worn out than other pages.

The law has been used as a method to detect fraud; for example, an accountant could detect a fraudulent tax document if the occurrence of leading digits does not sync up closely with Benford's Law.