The First-Digit Law

A Coruña	243,870
Ávila	58,358
Barcelona	1,604,555
Bilbao	345,141
Cáceres	95,617
Ciudad Real	74,427
Córdoba	327,362
Cuenca	55,428
Donostia	186,095
Girona	97,586
Granada	235,800
Guadalajara	83,391
Huelva	146,318
Huesca	52,239
Lugo	98,134
Madrid	3,141,991
Málaga	569,130
Murcia	439,889
Ourense	106,231
Oviedo	221,870

This post is about long lists of — real-life — numerical data. Doesn't sound like we're off to an interesting start, are we? But keep staring at such a list long enough, and you'll notice something strange.

For example, take a list of population counts for each of the more than 8000 cities in Spain. Like the one on the left. Without actually seeing that whole list, approximately what percentage of those numbers do you expect to start with the digit one?

You could reason that since the first digit could be any from 1 to 9, then a number will start with a specific digit about one in nine times as well. Or about ~11%. You know, on average.

That seems a reasonable and logical guess, and yet it turns out that almost one third of the numbers on this list start with the digit ‘1’!

Check out that link. That site shows something else too. It doesn't just apply to this particular list of numbers. Nor does it only apply to lists of population counts. There are many, many numerical data sets out there where this Law — also called Benford's Law — applies.

Socio-economic data. Stock prices. The lengths of rivers in miles. The lengths of rivers in kilometers! Street addresses. Constants in physics. Birth rates. Death rates. The sizes of the files on your computer. It doesn't apply to everything, but it sure applies to a lot.