The missing bullet holes and Abraham Wald

Chris Dobbert
4 min readMar 4, 2019


During WWII, the Navy tried to determine where they needed to armor their aircraft to ensure they came back home. They ran an analysis of where planes had been shot up, and came up with this.

The Statistical Research Group (SRG), where Wald spent much of World War II, was a classified program that yoked the assembled might of American statisticians to the war effort — something like the Manhattan Project, except the weapons being developed were equations, not explosives. And the SRG was actually in Manhattan, at 401 West 118th Street in Morningside Heights, just a block away from Columbia University. The building now houses Columbia faculty apartments and some doctor’s offices, but in 1943 it was the buzzing, sparking nerve center of wartime math. At the Applied Mathematics Group−Columbia, dozens of young women bent over Marchant desktop calculators were calculating formulas for the optimal curve a fighter should trace out through the air in order to keep an enemy plane in its gunsights. In another apartment, a team of researchers from Princeton was developing protocols for strategic bombing. And Columbia’s wing of the atom bomb project was right next door.

But the SRG was the most high-powered, and ultimately the most influential, of any of these groups. The atmosphere combined the intellectual openness and intensity of an academic department with the shared sense of purpose that comes only with high stakes. “When we made recommendations,” W. Allen Wallis, the director, wrote, “frequently things happened. Fighter planes entered combat with their machine guns loaded according to Jack Wolfowitz’s (Paul’s dad)recommendations about mixing 5 types of ammunition, and maybe the pilots came back or maybe they didn’t. Navy planes launched rockets whose propellants had been accepted by Abe Girshick’s sampling-inspection plans, and maybe the rockets exploded and destroyed our own planes and pilots or maybe they destroyed the target.”

So here’s the question:

You don’t want your planes to get shot down by enemy fighters, so you armor them. But armor makes the plane heavier, and heavier planes are less manoeuvrable and use more fuel.

Armoring the planes too much is a problem; armoring the planes too little is a problem. Somewhere in between there’s an optimum.

The military came to the SRG with some data they thought might be useful. When American planes came back from engagements over Europe, they were covered in bullet holes. But the damage wasn’t uniformly distributed across the aircraft. There were more bullet holes in the fuselage, not so many in the engines.

The officers saw an opportunity for efficiency; you can get the same protection with less armor if you concentrate the armor on the places with the greatest need, where the planes are getting hit the most. But exactly how much more armor belonged on those parts of the plane? That was the answer they came to Wald for. It wasn’t the answer they got.

The armor, said Wald, doesn’t go where the bullet holes are. It goes where the bullet holes aren’t: on the engines.

Abraham Wald, a statistician, disagreed. He thought they should better armor the nose area, engines, and mid-body. Which was crazy, of course. That’s not where the planes were getting shot.
Except Mr. Wald realized what the others didn’t. The planes were getting shot there too, but they weren’t making it home.

Wald’s insight was simply to ask: where are the missing holes? The ones that would have been all over the engine casing, if the damage had been spread equally all over the plane? Wald was pretty sure he knew.

The missing bullet holes were on the missing planes. The reason planes were coming back with fewer hits to the engine is that planes that got hit in the engine weren’t coming back.

Whereas the large number of planes returning to base with a thoroughly Swiss-cheesed fuselage is pretty strong evidence that hits to the fuselage can (and therefore should) be tolerated. If you go to the recovery room at the hospital, you’ll see a lot more people with bullet holes in their legs than people with bullet holes in their chests. But that’s not because people don’t get shot in the chest; it’s because the people who get shot in the chest don’t recover.

What the Navy thought it had done was analyze where aircraft were suffering the most damage. What they had actually done was analyze where aircraft could suffer the most damage without catastrophic failure. All of the places that weren’t hit? Those planes had been shot there and crashed. They weren’t looking at the whole sample set, only the survivors.

Here are further readings:

Abraham Wald’s Work on Aircraft Survivability by M. Mangel & F.J. Samaniego

An excerpt from How Not To Be Wrong by Jordan Ellenberg