Magic Numbers

The other day I came across a very weird mathematical thing. It’s called Benford’s law, and it describes how numbers generated by more or less any process form themselves into the same pattern. Suppose you made a list of the populations of all the towns in Britain, or a list of the individual sums you paid for all the cameras you’ve ever bought. You might suppose that the first digit of each number in one of those lists would be equally likely to be anything from 1-9, but it turns out that is not so. With any large sample of numbers you find that about 30% of them begin with a 1, 18% with a 2, right down to 4.6% starting with a 9. This was first discovered in 1881, but it took until 1996 for a statistician to work out why. The reason is too complicated to go into here, and anyway need not trouble us.
It turns out, though, to have real practical applications. For example, it has been used by forensic accountants to root out fraud, where financial figures don’t follow Benford’s law, and millions of pounds worth of fraudulent expense claims or rigged sales figures have been detected. It can be used to detect any reasonably large set of data that does not follow the natural or expected pattern, having been subjected to some sort of manipulation.
So I thought, I know a set of data which has probably been manipulated more than most by manufacturers over the years, and that is the serial numbers they engrave on the items we buy. We know, don’t we, or at least we reckon, that manufacturers don’t start their counting at one, they leave gaps in their sequences, all sorts of tricks mostly aimed at inflating the apparent number of items sold. So as I have a little database in my computer of all the photo-historical things that I own or have owned, it was time for a little experiment.
It turned out that I had records of 655 serial numbers which start with digits (some of them start with letters and I left those out), relating to cameras, lenses, shutters, accessories and so on. That seems to be a large enough random sample to use Benford’s law to see if the world’s manufacturers of photographic goods do indeed cheat when it comes to setting serial numbers. And they do.
Looking at the first digits of my serial numbers, the digits 1, 2, 5, 6 & 7 do indeed turn up with a frequency reasonably close to what Benford’s law predicts. However, 8 shows up 16% too often, 4 is 18% too rare, 3 is 21% over, and 9 is an enormous 37% in excess. I don’t know how much you can make of the specific ups and downs, but it’s clear in particular that too many of these numbers start with 8 or 9.
So I have demonstrated statistically what we already knew, that over the years manufacturers have indeed used their commercial skills to tweak their serial numbers for their own ends!