The First Digit Law

November 21, 2006 at 12:57 am 6 comments

If I were to pick a random city in the world, and tell you its population, what might the first digit of that number be?

You may think there’s equal probability for the first digit to be 1 to 9, but over 30% of the time it’s 1 (one).

Why? Think about it this way: let’s say a stock price doubles every year, starting at $100/share; it would spend a year with a first digit of 1 until it reaches $200, a year as $2xx or $3xx until it reaches $400, a year as $4xx, $5xx, $6xx, or $7xx, and then just a month or so at $8xx or $9xx, and all of a sudden it’s at $1,000 and the first digit is 1 again. Now it takes a long time (a year) to reach $2,000. There is a disproportionate amount of time when the stock price begins with the digit 1.

Many things in nature increase logarithmically. Benford observed this first-digit phenomenon in places including populations, addresses, baseball statistics, area of rivers, specific heats of compounds, and death rates. This rule has been used to identify accounting fraud where made-up numbers don’t match the distribution found in real accounting numbers.

Benford sampled over 20,000 numbers, and noticed the distribution of numbers was as follows,The First Digit Law

Digit Occurence
1 30.6%
2 18.5%
3 12.4%
4 9.4%
5 8.0%
6 6.4%
7 5.1%
8 4.9%
9 4.7%

This can be closely modeled using the log distribution of

F_a = log(1 + 1/a)

where F_a is the frequency that the digit a is the first digit in used numbers.

Additionally, the frequency of the n-th digit of a number can also be calculated using a similar formula, presented in the paper.

This is the law of anomalous numbers. We’ve 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.

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6 Comments Add your own

  • 1. Alan McCornack  |  December 9, 2006 at 11:30 pm

    Very useful, I’ve read about this before. Thanks.

  • 2. Butt Licker Magee  |  January 21, 2007 at 11:19 am

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  • 4. Arnob Endry  |  July 28, 2017 at 11:35 pm

    Online gaming falls into that category. I believe in doing everything in moderation, but I can see how online gaming and such activities can be addictive and in some cases a way to take time from venturing out to fitness centers, parks, and other social gatherings.
    As long as people are not harming others,

  • 5. smart fashion buying house  |  July 28, 2017 at 11:36 pm

    I personally think that the games are not the issue here. The games are just a tool for weaker individuals to use a form of escape. I play games but I dont live by them. I only play unranked games and I play for recreation but I limit my use as its easy to get too involved. The problem is games get better and they are inteneded to make you feel like not much time has passed and this sucks you in. I know a person who now has realised he used games as an escape, but I think it goes deeper than that still and becomes more of an addiction with mmorpg’s and ranked online games because people love compitition. But one quote I heard from my friend is, life is better when I have my headset on I get respect people like my and my life is better, plus I’m not a n00b anymore and it makes it more fun. The new language has to be looked at and I also feel that people that play games so religiously should be helped and not shunned.

  • 6. fried rice  |  October 26, 2018 at 2:04 am

    Everything is very open with a clear description of the issues.

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