Most commonly appearing first digit in (large) lists of numbers

A while ago in a GQ thread, someone mentioned what I believe were two related theorems in statistics.

The first theorem was something along the lines of “if you have a list of items (such as city populations) ranked by order of magnitude, the kth item will be approximately 1/k[sup]th[/sup] of the largest”, eg. the third city on a list of citys’ populations will be approximately 1/3 that of the largest city.

The second theorem said that something along the lines of “in lists of numbers (such as street addresses or country’s GNP’s), more numbers will start with ‘1’ than ‘2’, and more ‘2s’ than ‘3s’ etc.”

I know this is pretty vague but hope you can point me to the link/thread.


From The Power of One

Benford’s Law at Mathworld

New Scientist ran an article on the subject in July 1999.

The first one is Zipf’s Law:


Thank you both.

By the way, neither of those is a theorem. The first might be an empirical rule of thumb, good for many situations, but it’s far from universal. The second-tallest man in the world, for instance, is not half the height of the tallest man in the world. Really, it’s just a statement that a certain sort of distribution is common in the real world.

The second might be the bare-bones nugget of a provable statement, but you would need a lot more specification about the distributions, etc., to make it proveable.

You are quite right, Chronos. I was being lazy by using the term “theorem”.

But teach me, though, please. Isn’t “Benford’s law”, as described in the Mathworld link above (which I don’t really understand) pretty close to a theorem?