Thursday, February 03, 2005
Cycled to the shop to get fresh milk for an afternoon cup of tea: still shut. On the way back, milkless, hit a bump which jarred my spine. Checked the back tyre: it looked a bit flat. Tried to pump the tyre up, having great difficultly setting the valve right to let air in and not straight back out. Got mad, made a cup of tea with tinned milk. Switched on the radio. They were discussing a phenomenon which I heard as Bansford's Law. Applied to accounting it is said to detect dodgy figures. Googled it to find it was Benford's law.
Benford's Law was so interesting it took the anger over being unable to pump my bicycle tyre and the displeasure at the taste of evaporated milk completely away.
Being no mathematician, I determined to find the simplest explanation possible. Benford's Law at Mathspages.com, seemed not to be much use to me since it was full of equations, but the Google lisiting had mentioned: "Although there have been many lengthy and erudite "explanations" of Benford's Law, it seems to me it can be explained with a single picture: ...", so I kept on reading to find the "easy" pictorial explanation:
a logarithmic scale of base 10 numbers 1-9, which alone would not do much explaining.
Actually, this helps as a reminder after a graph from which this has been simplified is understood.
Beford's Law part I - How to spot a tax fraud has two tables which do expalin it more clearly. This is the first:
Digit 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9
odd of obtaining as first digit (%)
30.1 / 17.6 / 12.5 / 9.7 / 7.9 / 6.7 / 5.8 / 5.1 / 4.6
The logarithmic scale can be drawn up by apportioning each number such that
all numbers starting with a "1" will occupy 30.1% of the total length of the scale. Numbers like 1.23784, 1.5, or 1.879 would fall in this region.
A mathematical dunce should by now have grasped the basics. Advisable to try reading the articles though: there are a variety of examples of the law's application, including how to check Bill Clinton's expenses and how to impress your students by distinguishing between genuine and made-up coin-tossing statistics.
The intriguing bit for me was Benford thinking this all up because he noticed that page 1 of logarithm books he used were the most thumbed. The law was independently discovered in 1881 by the American astronomer Simon Newcomb, who everyone ignored.
A short piece in Mathworld has some useful explanatory tables.