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COMPUTING SCIENCE

E Pluribus Confusion

There’s more than one way to turn census data into congressional seats

Barry Cipra

Click to Enlarge ImageEvery decade the United States body politic undergoes a spasm of a decidedly mathematical nature. It will do so again in the next couple of years. That’s when the 2010 census will be used to determine the face of the next Congress, in a computational process called apportionment. The computation boils down to figuring out how many seats each state receives in the House of Representatives. But that glosses over a host of subtleties—and with it, a history of political rancor.

It wouldn’t seem that hard: You take the census state by state, you settle on the size of your House, and you assign each state a number of seats in proportion to their percentage of the total population. The solution is expressible in a formula that a grade-schooler can grasp: s=Sp/P, where s is the number of seats a state with population p should receive in a House with S seats altogether for a nation with P people in all. Under current law, the formula is even simpler: Since 1964, the size of the House of Representatives has been fixed at 435, so s=435p/P.

The problem, of course, is that Sp/P is rarely a whole number. Apportionment is a matter of turning fractions into integers. You might think the Constitution would have specified a procedure for doing so. It doesn’t. The closest it comes to a formula for anything actually turns integers into fractions: Our founding fathers called for the census to count each slave as three-fifths of a person. The Consitution left it up to Congress itself to decide not only how big to be, but how to dole out the seats. Its only stipulations were that each state get at least one seat, and that there not be more than one representative for each 30,000 people. If you like formulas, that’s 1 ≤ s ≤ max(p/30000, 1).

The one-per-30,000 ceiling was a serious constraint early on. Indeed, it entered into the argument over the initial size of Congress. Nowadays it’s not an issue, with the smallest state, Wyoming, having nearly a half-million residents.




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