Benford's Law vs the World

Today I got the chance to bust out EXCEL and show off some moves that are incredibly interesting. A few months ago I started listening to Public Radio Radiolabs podcast. The first episode I listened to was called Numbers and it blew my mind (http://www.radiolab.org/2009/nov/30/). They were talking about how the way we interpret numbers is…Subjective. Yes, their argument is that simple mathematics is different from person to person 

Now that idea flies in the face of everything I have ever thought about the philosophy of mathematics. Numbers are indelible. They are truth. 

But Radiolab listeners know that everything I knew from youth is not true. People have developed the construct of numbers based on an interpretation of the world, based on day to day experiences. We learned to count a few things at a time. And this has structured our brains to think mostly in terms of a few things. We subjectively appreciate the importance of a few items over many items. Their argument is that people think about mathematics naturally in logarithmic space rather than arithmetic space. 

One of the proofs they use is the idea of Benford’s law. Benford’s law is a proof asserting that unconstrained, natural systems will always occur in numbers that favor the logarithmic scale. There will be more ones than twos than threes and very few eights and nines. It is an interesting proof, you should check it out. The link goes to a nice description. And you can prove it too. Go find a list of bank records or temperature measurements. Count the frequency of the number of the first digit in each record and talley them up, then find the percentage of the total. 

I did this today for a dataset, it summarizes the percentage of first digit numbers predicted by Benford’s Law and those measured from a publicly available database of natural measurements that shall remain unnamed. They are very similar. 
                   Predicted
                   Benfords                     Measurements
Ones           30.1                           27.9%
Two's          17.6                           17.2%
Three's       12.5                           11.4%
Fours          9.7                             8.9%
Fives          7.9                              9.5%
Sixes          6.7                              6.6%
Sevens       5.8                              5.5%
Eights         5.1                              6.9%
Nines         4.6                               6.0%

This was a huge surprise to me. Before today, I was amazed by the story of Bedford’s law, but this is the first time it became real and I could prove it. Very Cool. 

Looking out for number one | plus.maths.org

plus.maths.org

So, here's a challenge. Go and look up some numbers. A whole variety of naturally-occuring numbers will do. Try the lengths of some of the world's rivers, or the cost of gas bills in Moldova; try the population sizes in Peruvian provinces, or even the figures in Bill Clinton's tax return. Then, when