What are the basic math skills that everyone should know?
Nipun Ramakrishnan | 丁仔 译
In the heat of World War II, a lot of American planes were coming back riddledwith bullet holes. One of the interesting things about this situation was that thebullet holes were concentrated in different regions of the plane. Some planescame back with bullet holes in the engine, some in the fuselage, etc.
The military saw an opportunity for efficiency. They realized that if they couldfocus on fortifying the areas of the plane that were the most prone to being hitand seriously damaged, they could not only save more planes, but also save onthe amount of armor they use. The military came to Abraham Wald and a fewother leading statistical researchers with the following data on the survivingplanes and where the bullet holes were concentrated:
于是，军队发现了提高效率的可能性。他们认为，如果着重强化飞机的几个容易被打击或者容易损坏区域，那么不仅能够拯救更多的飞机，而且还能节省装甲材料（抵御敌火的材料）的用量。军方找到了 Abraham Wald （一位数学家）还有几个顶尖的数据分析师，给他们提供了幸存飞机的数据，上面有弹坑分布的区域。
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 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:
1. Bullets just happened to hit every other part of the plane more often than ithit the engine
2. The engine is a point of vulnerability
And it doesn’t take a genius to see that the second explanation is a lot morereasonable than the first and after Wald’s recommendations were put intoeffect, a lot more American planes were saved.
一般人都可以看得出，第二个解释比第一个更合理。听取了 Wald 的建议重新改造之后，有更多的美国飞机得以幸存。
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.
能够读出数据背后隐藏的前提，是我们常常低估的数学能力。Abraham Wald 能够解决这个问题是因为他意识到军方假设数据能够代表所有飞机的情况，而不只是幸存的飞机。今天，我们在很多地方看到的数据都是有误导性的，它试图呈现你们想看到的东西。如何区分事实和误导就取决于你能否读懂数据假设的前提了。
If you want to see more of these types of examples, I highly recommend Jordan Ellenberg’s How Not to Be Wrong: The Power of Mathematical Thinking which 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.
如果你想看更多此类的例子，我强烈建议你去读 Jordan Ellenberg 的 《如何不犯错：数学思维的力量》。书里面很好地讲述了 Wald 的例子，以及许多忽略数据假设的前提而得出荒谬结论的案例。