Math in Elections: Polls, Percentages, and Seats

I stayed up late once watching results come in, and by midnight the wrong side was winning. Not the side I wanted — the side with fewer votes. Party Y had 170 votes to Party X's 130, a clear thirteen-point margin, and Party X took two of the three seats. I assumed I had misread the graphic. I had not. The votes were right. The seats were right. The connection between them was the part nobody on the broadcast explained.

That connection is the whole subject. An election is two calculations stacked on each other: first people vote, then votes become seats, and the second calculation follows rules that have almost nothing to do with the first. Percentages decide who is popular. A different piece of arithmetic decides who governs. The rest of this article runs both calculations by hand, with numbers small enough to check on paper, so the surprise stops feeling like a trick and starts feeling like a procedure.

The same votes, a different winner

Take three districts with 100 voters each, and give each district to whoever gets the most votes there. Here is one way the night can go:

• District A: X 60, Y 40 — seat goes to X
• District B: X 60, Y 40 — seat goes to X
• District C: X 10, Y 90 — seat goes to Y

Add them up. X has 130 votes, Y has 170. Y wins the popular vote 56.7% to 43.3%. X wins the seats 2 to 1. Both totals are correct at the same time, because the seat count never saw the vote totals — it only saw three separate winners.

The votes that did not change anything have a name: wasted votes. Y's extra 80 votes in District C elected nobody beyond the first seat, and X's losing 40s in A and B elected nobody at all. A system that awards whole seats per district throws away every vote above the winning line and every vote below it. That is not fraud and not a mistake. It is what winner-takes-all means, written out in numbers you can check.

Count them and the waste is startling. In a 100-voter district, 51 votes win, so everything past 51 is surplus and everything on the losing side is gone. District A wastes 9 of X's surplus plus all 40 of Y's. District B wastes the same 49 again. District C wastes 39 of Y's surplus plus all 10 of X's. Total wasted: 28 of X's 130, and 119 of Y's 170. Nearly half the votes cast — 147 out of 300 — elected nobody. Y won the night and still watched two-thirds of its own votes do nothing.

Dividing five seats among four parties

Other systems try to waste less. Suppose 100 voters elect 5 seats, and the votes split A 46, B 29, C 15, D 10. Five seats from 100 votes means each seat "costs" 20 votes — that number is called the quota. Give every party as many whole quotas as it earned, then hand the leftover seats to the largest remainders:

Three seats go out on whole quotas, and the last two go to the two largest remainders, C's 15 and D's 10. Final result: A 2, B 1, C 1, D 1. This procedure is called the largest remainder method, and several countries use it for exactly this reason — it keeps the seat shares close to the vote shares instead of discarding the leftovers.

Notice what the table really is: a division problem with sharing rules, the same shape as splitting anything fairly. Our division as sharing guide covers the underlying idea, and the quota-and-remainder step is why our rounding guide matters outside the classroom — here, how you round decides who governs.

Eleven votes flip the map

Now move eleven voters in District A from X to Y. The district goes from 60–40 to 49–51, and the seat flips. The new totals are X 119, Y 181 — Y still wins the popular vote by an even bigger margin, but the seats go from 2–1 for X to 2–1 for Y. Eleven people out of three hundred changed who governs, while the other 289 watched their votes keep meaning exactly what they meant before.

This is why close districts get all the attention and safe ones get ignored. A vote in a 60–40 district is worth almost nothing at the margin; the same vote in a 51–49 district can be the whole election. Campaigns know this arithmetic cold, which is why they spend where the map is close and not where the voters are many. The map, not the mood of the country, decides where the persuasion goes.

What a poll can and cannot tell you

This is where polls get misread. A poll measures votes: who people say they will choose, with all the sampling problems that come with asking. How a sample can go wrong — who is asked, who answers, and what the margin actually means — belongs to our surveys and sampling guide, which works through it properly.

Seats are a second calculation stacked on top of the first. A poll that says 52% to 48% with a three-point margin is telling you the vote share is somewhere near even. It is telling you nothing about the seats until you run those votes through the district map or the allocation rule. Two elections with identical polls can produce opposite parliaments, because the poll never contained the second calculation. Percentages describe popularity, and our percentages guide covers that half. Seats describe a procedure applied to those percentages, and the procedure is where the surprises live. Next time a broadcast shows a vote share and a seat count side by side, you will know which one was measured and which one was computed — and why they are allowed to disagree.