German tank problem

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The German tank problem came out in World War II. The Germans numbered their tanks by the integers (i.e. 1, 2, 3, ..., n). The Western Allies wanted to estimate the number of tanks the Germans had.^[1] To do this, the Allies devised a formula:

$m + \frac{m}{n}$

where m is the largest serial number observed and n is the number of tanks observed.^[2]

1 Example
2 Derivation
3 See also
4 References

[edit] Example

Say there are 15 tanks, numbered 1, 2, 3, ..., 15. You are an intelligence officer, and you have spotted Tanks 2, 6, 7, 14. By the above formula, m = 14, n = 4 and the formula gives 17.5, which is pretty close to the actual number of tanks, 15. Say if you spot Tanks 4 and 8 now. Now n = 6 and m still is 14. The formula gives a better estimation of 16.333..

[edit] Derivation

The formula can be interpreted as finding the average size of the gap (given by m/n) and then adding that to the highest tank, giving a fairly good estimation. This formula is simple and intuitive.

[edit] See also

Order statistics

[edit] References

^ Gavyn Davies How a statistical formula won the war The Guardian, 20 July 2006
^ Joyce Smart. German Tank Problem Logan High School cites Activity Based Statistics [by Richard L. Scheaffer (?)] p. 148-150. Exploring Surveys and Information from Samples, [by James M. Landwehr (?)] Section IX, p. 75-83. Statistical Reasoning, Gary Smith, p. 148-149