## The law of the first digit

*November 21, 2006 at 12:57 pm*

If I randomly picked a city in the world and gave you its population, what could the first digit of that number be?

You might think the first digit is equal to 1 through 9, but over 30% of the time, it’s 1 (one).

Why? Think of it this way: let’s say a share price *Double* every year starting at $ 100 / share; it would spend a year with a first digit of 1 until it reached $ 200, a year as 2xx or 3xx $ until it reached 400, a year as 4xx $, 5xx $, 6xx $, or 7xx $, and then just one Month or so at $ 8xx or $ 9xx, and suddenly it’s $ 1,000 and the first digit is 1. Now it takes a long time (a year) to get to $ 2,000. It takes a disproportionate amount of time if the share price starts with the number 1.

Much in nature increases logarithmically. Benford observed this phenomenon in the first digit in locations including populations, addresses, baseball statistics, river areas, specific temperatures of compounds, and death rates. This rule was used to detect accounting fraud, in which made up numbers do not match the distribution in real accounting numbers.

Benford selected over 20,000 numbers and found that the numbers are distributed as follows:

Digit | Happen |
---|---|

1 | 30.6% |

2 | 18.5% |

3 | 12.4% |

4th | 9.4% |

5 | 8.0% |

6th | 6.4% |

7th | 5.1% |

8th | 4.9% |

9 | 4.7% |

This can be modeled exactly using the logarithmic distribution of

**F_a = log (1 + 1 / a)**

where F_a is the frequency with which the digit **one** is the first digit in numbers used.

In addition, the frequency of the nth digit of a number can also be calculated using a similar formula presented in the work.

This is the law of anomalous numbers. We have learned to count 1, 2, 3, 4, … but nature counts 1, 2, 4, 8, …

Benford, F. (1938). The law of anomalous numbers. *Proceedings of the American Philosophical Society*, 78 (4), 551-572.

Entry filed under: 1, digits, math, numbers, statistics.