# Benford’s Law

##### The times they are a-changin’.

This post seems to be older than 15 years—a long time on the internet. It might be outdated.

There is been some interesting discusion of Benford’s Law in the last few weeks, first on the Internet, then in an episode of Numb3rs, then with my mom and grandma, and now back on the Internet again.

Benford’s Law…states that in lists of numbers from many real-life sources of data, the leading digit 1 occurs much more often than the others (namely about 30% of the time). Furthermore, the larger the digit, the less likely it is to occur as the leading digit of a number. This applies to figures related to the natural world or of social significance; be it numbers taken from electricity bills, newspaper articles, street addresses, stock prices, population numbers, death rates, areas or lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature).

One should occur 30.1% of the time, 2: 17.6%, and so on as the following list shows:

1. 30.1 %
2. 17.6 %
3. 12.5 %
4. 9.7 %
5. 7.9 %
6. 6.7 %
7. 5.8 %
8. 5.1 %
9. 4.6 %

Even more interesting, at least in my mind, is that Benford’s Law is easily formulated:
Logb(n + 1) – Logbn; where n is the number whose probability you are trying to determine and b is the base of the counting system (i.e. Base 10 for most purposes, Base 2 for binary, Base 16 for hexidecimal, etc).

There are many intersting things you can do with Benford’s Law, such as catching cheaters. Apparently, the IRS uses Benford’s Law as a preliminary check of your tax returns.

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## 5 thoughts on “Benford’s Law”

1. seems ridiculous to me too.

the only explanation I can fathom right now is rounding and truncation

2. It does seem rather incredulous, but I assure you that it is a proven phenonmenom. Wikipedia states, “This [Benford’s Law] can be explained by the fact that if the first digits have a particular distribution, it must be independent of the measuring system used. Specifically, this means that if one converts from e.g. feet to meters (multiplication by a constant), the distribution is unchanged ? it is scale invariant, and thus logarithmic.”

3. Ferg, incredulous means skeptical. You can be incredulous, but Benford’s Law can’t.

4. Right you are.
Benford’s Law is incredible and you are incredulous of it.