James Allison
Professor Emeritus, Psychology, Indiana University

The indisputable sign of a partisan gerrymander is the track it leaves behind in an election. This track consists of material evidence that the worth of one’s vote depended on one’s political party—a clear violation of the constitutional principle that one person’s vote is worth the same as any other’s. But how can we measure the value of one’s vote?

At the service station we can measure the gasoline value of our dollar as the number of gallons we get per dollar. I suggest that at the ballot box we can measure the congressional value of our vote as the number of seats we get per vote. If my candidate wins, my share of that one seat is 1/N, where N is the number of us who voted for that candidate. Under our winner-take-all system, If our candidate loses we get nothing, so each of our loser shares is worth 0/N seats.

What can our seats/vote measure tell us about a real congressional election? For example, in 2018 North Carolina elected 13 candidates to the U.S. House of Representatives under circumstances that reeked of a partisan gerrymander. One such circumstance was that the Democratic party averaged 47% of the popular vote across the 13 congressional districts, but won only 23% of the seats (3/13), while Republicans averaged 52% of the popular vote but took a whopping 77% of the seats (10/13).

How can this be, when the difference between their popular votes, 47% vs. 52%, was statistically imperceptible?¹ If all votes were worth the same, why didn’t Democrats win 6 of those 13 seats (47% of 13 = 6), and Republicans only 7 (52% of 13 = 7)? Does our seats/vote metric tell us that in North Carolina, in 2018, a Republican vote was worth more than a Democratic vote? The indisputable sign of a partisan gerrymander?

The short answer is “Yes!” Across the 13 congressional districts Democrats averaged .00000109 seats/vote. Republicans averaged significantly more, over four times that number, .00000483.² The same measure, derived from the 2018 elections in my state of Indiana, revealed significant Republican gerrymanders in the races for the U.S. House of Representatives and both houses of the state legislature. Those results were accompanied by legislative representation disproportionate to the popular vote.

For contrast consider Washington state, where in 2018 the two parties vied for eight seats in Congress.³ Here the legislative representation appeared more nearly proportional to the popular vote, as Democrats, with 56% of the popular vote won 5 of the 8 seats, and Republicans, with 44%, won 3. Accordingly, there was no significant difference in seats/vote, .00000274 for Democrats, and .00000237 for Republicans.⁴

These results suggest that the seats/vote metric can be used as evidence for or against the presence of a partisan gerrymander, as in North Carolina or Washington respectively. In its role as a standard, it can also reveal an apparent gerrymander as something less than significant. A case in point is Oregon’s 2018 election of four congressional candidates. Democrats, with 51% of the popular vote, won 75% of the seats (3/4), while Republicans, with 47%, won only 25% of the seats (1/4). Such a big departure from proportional representation might suggest the work of a partisan gerrymander, but what does our seats/vote metric say? Across Oregon’s four districts Democrats averaged .00000435 seats/ vote, nearly 3 times as many as Republicans’ .00000159, but this difference was not statistically significant.⁵ No gerrymander here.

Sticklers for reform might work to rectify that difference anyway, and if they did, the seats/vote metric could provide a useful standard of progress. Others might reason that a statistically insignificant difference is a difference not worth rectification. Such disputes are beyond the domain of inferential statistics. But we should keep in mind that when samples are relatively small—here, only four races for Congress—it can take a humongous difference to register as a statistically significant one.

We must bear in mind that the hint of a partisan gerrymander is not the same as material evidence of its presence. An example is Oregon’s false alarm, its apparent deviation from proportional representation. Minnesota reinforces that lesson with a bigger sample, 8 congressional races in 2018. In Minnesota Democrats averaged 55% of the popular vote and took 62.5% of the seats (5/8), while Republicans, averaging 44%, took only 37.5% of the seats (3/8). But was this disproportionate representation statistically significant? Not according to our seats/vote metric, where Democrats averaged .00000322 and Republicans .00000229—not a significant difference.⁶ No gerrymander here.

Both the tracking and the taming of the partisan gerrymander come together nicely in the example of Maryland’s eight congressional races in 2018. Democrats, who averaged 65.7% of the popular vote, won 87.5% of the eight seats (7/8), while Republicans, with 32.0%, won only 12.5% of those seats (1/8). The seats/vote metric shows that this was a genuine partisan gerrymander in favor of Democrats. Democrats averaged 00000450 seats/vote, significantly more than Republicans, .0000068.⁷ Now for the taming: How did this happen, and how might it be deterred?

It happened in the previous redistricting, when the party in control of the district mapping packed a huge number of Republican voters into Maryland’s District 1. As a result, the 183,662 Republican voters overwhelmed the 116, 631 Democrat voters in District 1, and thereby wasted 67,030 votes in the course of winning that race. They could have won it with only 116,632 votes. We could tame that gerrymander by unpacking District 1, say by moving 60,000 of its Republican voters over to District 6, adding them to the 105,209 Republican voters already there. That would give Republicans one more win, and Democrats one more loss, by a margin of 1,863 votes (165,209 to 163,346). Democrats would still have 65.7% of the popular vote, but now only 75% of the seats in Congress; Republicans would still have 32.0% of the popular vote, but 25% of the seats in Congress. And now there would be no significant difference between the two parties in terms of their average seats/vote, .00000374 for Democrats, and .00000177 for Republicans.⁸ No more gerrymander. The 60,000 Republican voters switched from District 1 to District 6 may seem like a lot, but amounts to only 3% of all voters.

And that is how to track and tame a partisan gerrymander. Anyone can do it, given the election returns and an elementary knowledge of inferential statistics.⁹

¹t(24) = 0.70, p = 0.49.

²t(24) = 3.85, p = .0008.

³In one additional race one of the two candidates ran as a Libertarian; in another both candidates ran as Democrats, an idiosyncrasy of Washington politics. These two contests were omitted from the analysis.

⁴t(14) = 0.25, p = 0.81.

⁵t(6) = 1.29, p = 0.25

⁶t(14) = 0.62, p = 0.54.

⁷t(14) = 3.99, p = .0013.

⁸t(14) = 1.37, p = 0.19.

⁹An elementary knowledge of inferential statistics would include an understanding of rules for deciding whether a difference is statistically significant. All of the present examples ask the same question, whether Republican and Democratic measures of seats/vote differ significantly. More precisely, what is the probability that the Republican and Democratic samples could have come from the same parent population of measures rather than two different populations? The statistical test—here, a test called “Student’s t”—produces a probability, p, that the Republican and Democratic samples came from the same parent population rather than two different populations. The scientific convention is that if p is equal to or less than .05—5 or fewer chances in 100—we reject the hypothesis that they came from the same parent population. We conclude instead that their difference was statistically significant. For further discussion and a convenient program for the automatic calculation of the t test, see graphpad.com.