German Tank Problem

Math teachers have been using the “German Tank Problem” for a while to teach estimators. It goes something like this.

The Allies capture five German tanks. Suppose that the serial numbers on the tanks are 15, 23, 59, 83, and 109. Provide an estimate of the number of tanks that were produced.

You can see that your estimate would be a great deal lower than the estimate if the serial numbers were 1015, 2394, and 9438.

The story and accompaning math problem was written about in the Guardian. My favorite line, which turned out to be true, was this:

The statisticians believed that the Germans, being Germans, had logically numbered their tanks in the order in which they were produced.

It turns out that the statisticians were spot on:

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.

The estimator is:

The best estimator to my first problem where the five serial numbers are 15, 23, 59, 83, and 109 would be:

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