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Proportionate ResponseDoes one Israeli really equal 47 Americans?

By Jordan Ellenberg
Posted Monday, July 24, 2006, at 12:29 PM ET

On June 25, Louise Roug and Doug Smith of the Los Angeles Times wrote that "[a]t least 50,000 Iraqis have died violently since the 2003 U.S.-led invasion. … Proportionately, it is equivalent to 570,000 Americans being killed nationwide in the last three years." Three weeks later on Meet the Press, Newt Gingrich asked viewers to "[i]magine Miami had missiles being fired at it every day. Remember that when Israel loses eight people because of the difference in population, it's the equivalent of losing almost 500 Americans."

It's hard for Americans to comprehend what's happening in the Middle East. That's why commentators reach for analogies. What event in the United States would be "equivalent" to the terror over there? The answer depends on what you mean by "equivalent." Is it, "What crime in America is morally equal to the killing of eight Israelis?" Or do you mean something more like, "What event would have the impact on America that the killing of eight Israelis does on Israel?"

The first question is easier. Unless you truly think Israeli lives are worth more or less than our own, the crime that's equivalent to the murder of eight Israelis is the murder of eight Americans. But what Roug, Smith, and Gingrich have in mind is something closer to the latter question. Their equivalences are shorthand for something like "an attack killing X percent of Iraq's population has the same impact as an attack killing X percent of America's population."

That description makes it easy to see how the factoids above were derived. The Times writers took the number of Iraqis who have died violently since the American invasion (50,000) and divided it by the population of Iraq (about 26 million) to find that 0.19 percent of Iraqis have been killed since 2003. The equivalent percentage of the American population, then, is 570,000. Gingrich's calculation works the same way. There are 6.3 million Israelis. Gingrich worked out that one in 787,500, or 0.00013 percent of Israel's population, was killed in the Hezbollah rocket attack. Multiply that proportion by the population of the United States, and you get something close to the quoted figure. (The correct number is closer to 380 Americans than 500, but Newt was in the ballpark.)

Do these calculations make any sense? In math, if you're not certain your argument is correct, you should apply it in some novel situations and see if it spits out reasonable results. For example: I live in Madison, Wis., which has a population of 220,000. If I want to compute what the Iraq war would look like in my town, I can just take 0.19 percent of that figure and find that the "equivalent" death toll here is 400 people. As for the Hezbollah attack, 0.00013 percent of 220,000 yields 0.286. So, Hezbollah's rocket attack killed the equivalent of a quarter of a Madisonian—surely an underestimate of the crime's impact. On the other hand, nothing forces us to use the whole population of Israel. The city of Haifa has a population of about 270,000, so 0.003 percent of its population was killed in the attack Gingrich cites. The equivalent in Miami (population 362,470) would be 11 people. So, what should we imagine in Miami? Eleven people dead, or Gingrich's 500? The method of proportions gives both answers—not a great advertisement for the technique.

There are cases where the method of proportions is a useful one. Human Rights Watch estimates that 75 percent of the Tutsi population of Rwanda was murdered in the 1994 genocide. It's hard to get a mental picture of such a crime—but imagining three-quarters of your neighbors hauled off to slaughter isn't a bad place to start.

Why does it make sense to use proportions when talking about Rwanda but not when talking about Haifa? The difference between the two cases is simple—75 percent is a big number, and 0.00013 percent is a really small one. You can imagine what it means to lose three-fourths of your population, but your innate number sense simply doesn't extend to "one in 787,500." Try to follow Gingrich's instructions and imagine the impact on a society in which one in 787,500 people had just been murdered. Now imagine that it was actually one in 78,750. You just multiplied the scale of the crime tenfold, but can you say sincerely that the two numbers inspire a different reaction? The proportions are just too small to comprehend.

How small a proportion is too small? Here's a good rule of thumb: If it makes sense to talk about the "survivors" of an event—as in the "Tutsi survivors of the Rwandan genocide"—it's probably reasonable to scale by population. (The Rwandan genocide would be the equivalent, in this sense, of killing 165,000 people in the city of Madison.) But only the most tendentious cranks would refer to "those Iraqis who survived the U.S. occupation," let alone call an Israeli living far from the war zone "a survivor of Sunday's Hezbollah rocket attack." That's your signal not to make the claim that each one of those Israelis is "equivalent" to 47 Americans, or four Iraqis, or one twenty-eighth of a Madisonian.

If you wanted to, you could incorporate the above reasoning into a more complicated formula. That formula would work one way in Rwanda, where most of the targeted population was killed, and another way in Iraq, where relatively few have died. But such a formula would lack the simplicity of the straightforward proportional argument, and complicated formulas have a way of lending undeserved authority to assertions—"the aftereffect of the Iraq War is the equivalent of 570,000 dead Americans"—that are supposed to be mere analogies. Maybe it's best to keep it simple. If you want to imagine eight people killed, imagine eight people killed—but people on your block, not across the world. That computation is mathematically and morally unimpeachable, and no calculator is required.

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Jordan Ellenberg is an associate professor of mathematics at the University of Wisconsin. His first novel is The Grasshopper King.
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