Fun with maths

[dieseltaylor]

I thought this cracking

How a statistical formula won the war

• The Guardian, Thursday 20 July 2006

Here is a story about mathematical deduction that I love, mainly because it is said to be true, and because it had an impact (albeit small) on the outcome of the second world war. It is the story of how a simple statistical formula successfully estimated the number of tanks the enemy was producing, at a time when this could not be directly observed by the allied spy network.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.

One solution was to ask intelligence to guess the number by secretly observing the output of German factories, or by trying to count tanks on the battlefield. Both the British and the Americans tried this, but they found that the estimates returned by intelligence were contradictory and unreliable. Therefore they asked statistical intelligence to see whether the accuracy of the estimates could be improved.

The statisticians had one key piece of information, which was the serial numbers on captured mark V tanks. The statisticians believed that the Germans, being Germans, had logically numbered their tanks in the order in which they were produced. And this deduction turned out to be right. It was enough to enable them to make an estimate of the total number of tanks that had been produced up to any given moment.

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

By using this formula, statisticians reportedly estimated that the Germans produced 246 tanks per month between June 1940 and September 1942. At that time, standard intelligence estimates had believed the number was far, far higher, at around 1,400. After the war, the allies captured German production records, showing that the true number of tanks produced in those three years was 245 per month, almost exactly what the statisticians had calculated, and less than one fifth of what standard intelligence had thought likely.

Emboldened, the allies attacked the western front in 1944 and overcame the Panzers on their way to Berlin. And so it was that statisticians won the war - in their own estimation, at any rate.

[Der Alte Fritz]

I like that story a lot - it has a simplicity about it that is appealing.

The only similar story that I have is from the C15th. The main dining hall at New College Oxford has a huge timber roof one of the largest in Europe and built in the C15th. At the start of the C20th it was noticed that the roof which was oak, was getting woodworm and would have to be replaced. But where asked the Dean and Chapter are we to get such great timbers to replace the roof? Trees had been heavily felled during the Napoleonic War and the Industrial Revolution, so there were no longer any great oaks growing. A Junior Dean spoke up fro the back and said why don't we ask the College Forester? The Forester was a gnarled figure who had no been seen in the college for 20 years. When he appeared he simply said "Oh I was wondering when you were going to ask about the Hall Copse". It appears that the original builders, knew full well that after 500 years an oak structure develops woodworm, being craftsmen of long standing. They also knew that it would take 500 years to grow trees of sufficient height to replace the roof, so being men of foresight, they planted a stand of oaks to replace it when the time came. This was known to the generations of College Foresters who had guarded the stand of trees through one national emergency after another.

Rarely do you find such an example of hope in the future as you do among the builders and Foresters of Oxford.

[Lt Bull]

Thank you dieseltaylor for posting that extraordinary equation.

I have been playing around with it and I have found that the following variation of it results in much more accurate results:

(M-1)(S+1)/S + 0.05*(M-1)(S+1)/S

I can't believe how accurate this equation is at predicting the maximum number in the series.

Of course it is assumed that the numbers in the sample are randomly chosen.