What are the basic math skills that everyone should know? by Nipun Ramakrishnan
Answer by Nipun Ramakrishnan:
In the heat of World War II, a lot of American planes were coming back riddled with bullet holes. One of the interesting things about this situation was that the bullet holes were concentrated in different regions of the plane. Some planes came back with bullet holes in the engine, some in the fuselage, etc.
The military saw an opportunity for efficiency. They realized that if they could focus on fortifying the areas of the plane that were the most prone to being hit and seriously damaged, they could not only save more planes, but also save on the amount of armor they use. The military came toand a few other leading statistical researchers with the following data on the surviving planes and where the bullet holes were concentrated:
Based on this data, which part of the plane do you think the military should fortify?
Most people would answer that it should be the fuselage, and they would be wrong.
The more accurate answer is the engine. Wald insightfully realized that the data that he had been given was not an appropriate representation of the problem. The reason that there were fewer hits on the engine is that the planes that got hit in the engine were not coming back. And the fact that a majority of the surviving planes had bullet holes in the fuselage is pretty strong evidence that the planes could tolerate damage to the fuselage.
Wald realized that there were only two explanations for the data:
- Bullets just happened to hit every other part of the plane more often than it hit the engine
- The engine is a point of vulnerability
And it doesn’t take a genius to see that the second explanation is a lot more reasonable than the first and after the Wald’s recommendations were put into effect, a lot more American planes were saved.
One of the most underrated mathematical skills is the ability to understand the underlying assumptions made by data. Abraham Wald was able to solve this problem because he realized the military made the assumption that the data represented all planes rather than just the surviving planes. A lot of the statistics that are thrown out in the world today tend to fool a lot of people into believing what they want to hear, but how you separate the facts from the bullshit is by understanding the assumptions made by the data.
If you want to see more of these types of examples, I highly recommend Jordan Ellenberg’swhich does a great job explaining the Wald example as well as many other examples in which forgetting about the assumptions behind the data can lead to ridiculous conclusions.