

(Photo from Flickr by PhotosNormandie. Some rights reserved.)

You'll have noticed the PDA blog's report about the latest estimates of how many iPhones have been sold - a solid 9.1m and counting.

But you might have wondered how the process of simply asking people for their phone's serial number, IMEI number and date of purchase would allow someone to make such a specific calculation of the number sold (which, according to the spreadsheet, was 9,190,680).

The answer: tanks. Well, the same process that allowed the Allies to calculate how many tanks the Nazis were producing during World War II. You can read Gavyn Davies's full-length account, or we'll give you the extracted version here.

It goes like this:

By 1941-42, the allies knew that US and even British tanks had been technically superior to German Panzer tanks in combat, but they were worried about the capabilities of the new marks IV and V. More troubling, they had really very little idea of how many tanks the enemy was capable of producing in a year. Without this information, they were unsure whether any invasion of the continent on the western front could succeed.

They could try spying on factories, or count tanks in the field. This was easier. Even better: get the serial numbers from the tanks, and estimate from this how many were being built.

The basic idea was that the highest serial number among the captured tanks could be used to calculate the overall total. The German tanks were numbered as follows: 1, 2, 3 ... N, where N was the desired total number of tanks produced. Imagine that they had captured five tanks, with serial numbers 20, 31, 43, 78 and 92. They now had a sample of five, with a maximum serial number of 92. Call the sample size S and the maximum serial number M. After some experimentation with other series, the statisticians reckoned that a good estimator of the number of tanks would probably be provided by the simple equation (M-1)(S+1)/S. In the example given, this translates to (92-1)(5+1)/5, which is equal to 109.2. Therefore the estimate of tanks produced at that time would be 109

The maths wonks reckoned the Germans were making 246 (precisely) tanks per month between June 1940 and September 1942.

Intelligence scoffed. Their number was much higher - those damn Germans were turning out 1,400 per month!

German production records captured after the war showed that there were actually 245 made per month. Score one to the maths wonks. (They then lost it all by betting on credit default swaps, but that's another story.)

The process works, basicaly, when you're trying to find out the production of a public item from a secretive organisation.

In fact if you really felt like it you could get people to pony up the serial numbers of their MacBooks/MacBook Pros/other Apple items, and it would probably help you to figure out Apple's production (depending, very much, on the size of the sample). Then again, it tells you how many it sells each quarter.

German tanks, iPhones.. what else needs calculating, do you think?