## German tanks and Whirlpool members

There’s a very interesting post up at the excellent Stats in the wild blog about how statisticians helped win the Second World War.

During World War II, the Allies were trying to estimate the number of a certain kind of German tank.  They needed this information to better plan their attacks and invasions.  There were two sets of estimates made, one by intelligence and another by a group using statistical methods.

The method that the statisticians used was quite simple in theory and turned out to be surprisingly accurate in reality.  It was based on the tanks’ serial numbers (assumed to be sequential), s, and the number of tanks observed, n.  I encourage you to read the Stats in the wild post for all the details, but essentially the formula used for estimating the true number of tanks in existence was:

(n+1)*max(s)/n

So, for example, if five German tanks were observed with serial numbers 7, 123, 99, 44, and 45 then the best guess for the total number of tanks would be (5+1)*123/5=148.

Simple, clever and, if history is a judge, highly effective.  I wondered if this method could be applied in a more modern setting.

I’m regular over at the Whirlpool Broadband Discussion Forums.  It seemed like an ideal testbed for the German tank counting theory.  When you become a member of Whirlpool you are assigned a User ID which, totally unsurprisingly, is a sequential number based on the order that you join.  I randomly chose a recently posted and publicly accessible forum thread where I observed 12 unique User IDs: 33561, 61589, 254627, 24658, 104905, 164131, 75498, 112974, 173866, 123400, 107050, and 99585.  The maximum of these observed values is 254627 so our estimate for the actual number of Whirlpool members is 13*254627/12=275,846.  This compares to the published size of (at the time of writing) “255,706 registered members”.  So the estimate is reasonably close to reality, overestimating by about 8%.

I’m really impressed with this method of estimating population sizes based on observed samples.  It’s gone straight into my “statistics toolbox”.  I thank Stats in the wild for writing about it.

I don’t know which would be more terrifying to meet in real life…  a Whirlpool member or the pointy end of a German tank.
Stanley Devia

“Can anyone recommend a good ISP?”