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Thinking outside of the box – Abraham Wald’s Memo

I love this story – it is the perfect example of thinking outside of the box! The following text is based on MacTutor History of Mathematics archive and Biographies.

Abraham Wald Background

Abraham Wald

Abraham Wald was a mathematician who contributed to decision theory, geometry, and econometrics, and founded the field of statistical sequential analysis.

Abraham Wald was born into a Jewish family in Hungary. It was a family of intellectuals but, being Jewish, they were forced to earn their living in trades well below their abilities. At this time both primary and secondary schools in Hungary required pupils to attend on Saturdays and the Wald family could not allow their son to attend school on the Jewish Sabbath. As a result Abraham did not attend either primary or secondary school and was educated at home by members of his family. This certainly did not put him at a disadvantage from an educational point of view for his family were very knowledgeable and competent teachers.

After World War I much of the land that had been part of Hungary was given to neighbouring countries and at that time Cluj became part of Romania. Wald was allowed to attend the University of Cluj but it appears that this was not made easy for him because he was Jewish. However his outstanding abilities in mathematics led him to wish to continue to undertake mathematical research and in 1927 he entered the University of Vienna to study with Karl Menger. He worked under Menger‘s supervision on geometry and was awarded his doctorate in 1931.



Abraham Wald’s Memo

His most famous contribution to statistics related to reviewing damaged aircrafts returning from Germany in the Second World War. Abraham found that the fuselage and fuel system of returned planes are much more likely to be damaged by bullets or flak than the engines.

Considering this information, the question was: What should he recommend to his superiors?

Wald’s brilliance suggested an unconventional solution which saved countless lives.

Don’t arm(our) the places that sustained the most damage on planes that came back – simply because, they came back and these areas can sustain damage!


This seems backward at first, but Wald realised his data came from bombers that survived. That is, the British were only able to analyse the bombers that returned to England; those that were shot down over enemy territory were not part of their sample. These bombers’ wounds showed where they could afford to be hit. Said another way, the undamaged areas on the survivors showed where the lost planes must have been hit because the planes hit in those areas did not return from their missions.

Wald assumed that the bullets were fired randomly, that no one could accurately aim for a particular part of the bomber. Instead they aimed in the general direction of the plane and sometimes got lucky. So, for example, if Wald saw that more bombers in his sample had bullet holes in the middle of the wings, he did not conclude that Nazis liked to aim for the middle of wings. He assumed that there must have been about as many bombers with bullet holes in every other part of the plane but that those with holes elsewhere were not part of his sample because they had been shot down.

Wald applied his statistical skills in World War II to the problem of bomber losses to enemy fire. So let us apply our statistical skills to improve the way we work and move society forward: Safer, Smarter, Greener

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