One of the Most Shocking Facts Ever Discovered: Benford's Law

One of the most freakishly astounding mathematical curiosities I have ever come across is known as Benford's Law, named for physicist Frank Benford, who stated it in 1938 (though Simon Newcomb first mentioned it back in 1881). This law is shocking and inexplicable, especially when you give it some consideration.


Newcomb was struck by something peculiar when he was checking through books filled with number tables. He noticed that the pages in the front of the books (containing sets of numbers beginning with 1) were far more worn than the later pages. From this, he realized that digits beginning with 1 occurred far more often than other digits.

However, it turns out that when you take lists of all sorts of numbers, drawn randomly from various sources, the first digit 1 appears with the greatest frequency, or 30.1% of the time. This is true for lists of street addresses, phone numbers, death rates, lengths of rivers, baseball stats, atomic weights, magazine articles, and millions of other data sources. This fact is freakish enough by itself. What makes Benford's Law so badass is that it turns out that the appearance of the other 8 digits (as the first digit in any number) fall in sequential order with lesser frequency, as follows: 

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%

How on earth do numbers "know" to appear with such predictable frequency? What is more, why would they appear less often in sequence? Why does 2 come in second place, rather than 6? Yet they show up in ordinal fashion 1 through 9. This insane finding makes no sense to most of us, who assume that numbers culled from Time Magazine or clothing sizes have to be random. Pythagoras would have been tickled by Benford's Law. He already believed the first 9 numbers were the source of all, with 1, the Monad, being a symbol of the Creator because all numbers derive from One and all numbers multiplied times One always equal One--the Master of All.

Take a random list such as the top 60 tallest buildings in the world. The digit 1 appears 43.3% of the time, followed predictably in descending order by the digits 2-9 (9 never appears in this case). The same findings result irrespective of the unit of measure (meters, feet, etc). 

The larger the set of numbers in a given list, the more closely each digit conforms to the law. There are of course number sets where the law does not apply, particularly if they are rigid (such as lists of human heights on average). However, if you were to take such lists and combine them with other data lists (the more the better), the law reappears again. 

There are numerous theories as to why digits follow this law, but of course no one really has a satisfactory answer.

Benfords' Law actually has some useful applications. Imagine, for example, a creative accountant cooking the books with what he or she feels are random numbers. Little does our future felon realize that the digit 8 should only appear about 5% of the time. The Law has been invoked in cases of election fraud, and often shows up in accounting and other financial fraud cases. In the United States, Benford's Law is legally admissible in court at all levels. 


Hey, Pythagoras, you were right: Number One is Number One!