### Benford’s Law and the Administrative Geography of Great Britain

Just listened to the latest episode of the Infinite Monkey Cage, and was reminded of Benford’s Law. This states:

Benford’s Law, also called theFirst-Digit Law, refers to the frequency distribution of digits in many (but not all) real-life sources of data. In this distribution, the number 1 occurs as the leading digit about 30% of the time, while larger numbers occur in that position less frequently: 9 as the first digit less than 5% of the time. Benford’s Law also concerns the expected distribution for digits beyond the first, which approach a uniform distribution.

I was curious if that might emerge in geography (or Ordnance Survey data) somehow. Turns out if we look at the areas (in metres squared) of the polygons in the Boundary Line Product (i.e. the areas of all the counties, wards, consistuencies, districts, parishes etc. in GB) then we get a pretty good fit. In the table below the first column is the leading digit of the polygon area, the second is the percentage of areas starting with that leading digit and the third column is the value Benford’s Law predicts:

1: 30.6 30.1

2: 15.9 17.6

3: 11.3 12.5

4: 9.8 9.7

5: 8 7.9

6: 7.3 6.7

7: 6.3 5.8

8: 5.6 5.1

9: 4.9 4.6