page_title: German tank problem
section_title: Similar analyses
hierarchical_section_title: German tank problem / Historical problem / Similar analyses
caption_attribution_description: English: V2-Rocket in the Peenemünde Museum Deutsch: V2-Rakete im Peenemünde Museum Français&#160;: Musée de Peenemünde. Nederlands: V2-raket in het Peenemünde Museum
context_page_description: In the statistical theory of estimation, the German tank problem consists of estimating the maximum of a discrete uniform distribution from sampling without replacement. In simple terms, suppose there exists an unknown number of items which are sequentially numbered from 1 to N. A random sample of these items is taken and their sequence numbers observed; the problem is to estimate N from these observed numbers.
  The problem can be approached using either frequentist inference or Bayesian inference, leading to different results. Estimating the population maximum based on a single sample yields divergent results, whereas estimation based on multiple samples is a practical estimation question whose answer is simple but not obvious.
  The problem is named after its historical application by Allied forces in World War II to the estimation of the monthly rate of German tank production from very limited data. This exploited the manufacturing practice of assigning and attaching ascending sequences of serial numbers to tank components, with some of the tanks eventually being captured in battle by Allied forces.
context_section_description: Similar serial-number analysis was used for other military equipment during World War II, most successfully for the V-2 rocket.
  Factory markings on Soviet military equipment were analyzed during the Korean War, and by German intelligence during World War II.
  In the 1980s, some Americans were given access to the production line of Israel's Merkava tanks. The production numbers were classified, but the tanks had serial numbers, allowing estimation of production.
  The formula has been used in non-military contexts, for example to estimate the number of Commodore 64 computers built, where the result (12.5 million) matches the low-end estimates.
caption: V-2 rocket production was accurately estimated by statistical methods
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