Benford’s Law

There may not be enough PPE in hospitals but popular economist, Tim Harford, has his own PPE from Brasenose. His columns in The FT and broadcasts on BBC Radio 4 are usually intriguing.

That’s how I heard about Benford’s Law. It is counter-intuitive and has practical applications so it’s worth knowing about. Wikipedia explains the law.

“Benford’s law, also called the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation about the frequency distribution of leading digits in many real-life sets of numerical data. The law states that in many naturally occurring collections of numbers, the leading significant digit is likely to be small. For example, in sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. If the digits were distributed uniformly, they would each occur about 11.1% of the time. Benford’s law also makes predictions about the distribution of second digits, third digits, digit combinations, and so on.”

There are plenty of examples of this curious law working, look at some of them here, but I want to try it out for myself. There are fifty motorways in the UK and they are ranked by length in miles and kilometres on Wikipedia. Here is my analysis of the incidence of first digits.

First Digit  Miles   Kilometres %

1                   24           26

2.                 28              6

3.                 12             20

4.                  8              18

5.                 12               8

6.                   2               6

7.                   6               2

8.                   6             10

9.                   2               4

The general point is made. There are more low first digits. I think it does not conform more closely to Benford’s Law because it is a small sample. It would be interesting to look, for example, at the FTSE 250 and subject share prices, capitalisation and turnover to the Law. But as it was laborious even compiling my motorway data so it’s not a job I’m up for.

The Law is more than a statistical quirk, it can be used to detect fraud. Tim Harford suggests that Greek macro-economic data some twenty years ago and GDP data from some African countries do not comply with the Law suggesting false reporting. Strangely Bernie Madoff’s investment returns do meet the Law’s expectations. My guess is that he extrapolated his returns from some genuine data.

A friend of mine lived for a while in a hotel in Colombo. My youngest goddaughter visited him as a child and found hotel life heavenly but my friend tired of being cooped up, know the feeling, and bought a house on the sea. A.S.H. Smythe is cooped up in Colombo and writes about how he is coping in The Spectator. He’d planned to climb Adam’s Peak ( 2,243 metres) with a friend; instead he has climbed it at home. There are seventy-two steps on his stairs so I reckon he lives in a big house. In nine days he had completed the climb. I only have thirty-six stairs and the risers are six inches so to climb a Munro will require 111 ascents. Assuming I go upstairs twice a day I climb  more than 2,000 feet every two months. No wonder the stair carpet is showing a bit of wear.

My boots are made for walking, I can’t go down town but this is a gem.