During WWII, statistician Abraham Wald was asked to help the British decide where to add armor to their bombers. After analyzing the records, he recommended adding more armor to the places where there was no damage!
This seems backward at first, but Wald realized his data came from bombers that survived. That is, the British were only able to analyze 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.
14 thoughts on “Selection bias and bombers”
So is the assumption of random firing justified?
Ling, I think the random firing assumption was justified at the time, given the state of technology. The standard error may have been larger than the size of an airplane.
Thanks, interesting story!
Can you cite the source of this excellent example?
Wald, Abraham. (1943). A Method of Estimating Plane Vulnerability Based on Damage of Survivors.
Here’s a reprint of one of his papers on the subject, and here’s another source.
However, while I’ve heard this story many time before, I had not heard it claimed that he did this for the British, but rather for the US Navy.
Thanks. This is good stuff as teaching material.
I thought that was beautiful, and something that should be a part of basic education for everybody. How many decisions are poorly made because of the problem of silent evidence? This plagues everything from our social lives to government policy, and as a species it’s about time we got over this.
Wait, so what were the final results? Did a higher %age of bombers come back after the fleet was fitted with the armor in Wald’s recommended areas?
Dan Meyer did a nice version of this for the high school crowd.
Really nice graphic he created to go with the material.
Cute statistical study but also suffers from randomness bias. Armor the most vulnerable points; fuel, controls and cockpit, engines etc. Then apply statistics to the rest if you have any weight budget left.
I believe the point of the study was to determine what the most vulnerable parts were.
I am commenting years later because this is so good. I only came across it more recently and now I cite it to people all the time. I am not yet at the point where I can intuitively spot the pattern in action, but I hope to get there. I reckon the world is full of people solving the most survivable problems and ignoring the problems that remove their victims from the sample.