Can Math Assist in Saving Democracy?

This is a sidebar to "Slaying the Partisan Gerrymander," which appears in the Fall 2017 issue of The American Prospect magazine. Subscribe here

When it comes to creating a test for detecting gerrymanders, getting stuck on the details of maps hinders an efficient evaluation. Courts need simple and straightforward tools for detecting gerrymanders.

Partisan gerrymandering is perpetrated using two complementary methods: cracking and packing. “Packing” occurs when as many supporters of one party as possible are crammed into a small number of districts, creating a few overwhelming wins for the victim party. The remaining members of the victim party are then “cracked,” or spread evenly across a large number of districts that the gerrymandering party can dominate. Fortunately, cracking and packing create a distinctive statistical pattern that can be detected with the help of a little math.

Two tests, Student’s t-test and the mean-median difference, probe whether a districting scheme is likely to have arisen deliberately.

In North Carolina, average winning margins for Democrats and Republicans differ systematically, consistent with a Democratic gerrymander until 2010 and a Republican gerrymander starting in 2012. For the averages to have arisen incidentally from a party-neutral process, they would have to both fall within the zone of chance (gray shaded region). 

Student’s t-test. The simplest way of detecting many partisan gerrymanders is to ask whether one side’s average wins are more lopsided than the other’s. Student’s t-test—the simplest statistical test in all the sciences—asks whether two averages differ more than would be expected by innocuous or chance events. In the case of redistricting, the t-test can identify when one side’s wins are unusually lopsided compared with the other’s, a sign that their voters have been packed.

The mean-median difference. An equally simple test is the difference between the average (or mean) and median vote share for all the districts in the state. In a closely divided state, gerrymandering creates an artificially high number of narrow wins by one party, thus lowering the party’s median vote share compared with its statewide vote share. A large difference between the mean and median (which is unlikely to arise by chance) is indicative of a partisan gerrymander. The mean-median difference can be calculated easily on pencil and paper by a judge or clerk in the margin of a brief.

Nearly all the partisan gerrymanders in this article fail either t-test or the mean-median difference test. The exception is Maryland, which is dominated by one party and has only one remaining Republican district, conditions that make statistical testing harder. In this case, a better approach is careful examination of what was done to convert the eliminated Republican district to a newly made Democratic district.


TWO OTHER TOOLS—Monte Carlo simulation and the efficiency gap—are available to measure the distortions in representation that arise as a consequence of partisan gerrymandering.

Monte Carlo simulation: A common tool in statistical inference, the Monte Carlo method is a sophisticated version of pulling numbers out of a hat. The question is simple: Given the statewide vote earned by one party, how many congressional seats would that party win given nationwide trends? The deviation in seats can be calculated by comparing the actual number of seats the party wins in that state to comparable elections across the United States. A large deviation suggests mischief may have been at work.

The efficiency gap: The efficiency gap was proposed in 2013 by Eric McGhee as a measure of the degree of a gerrymandering offense by measuring “wasted votes.” To win a district, a party needs just over 50 percent of the vote. Any votes beyond the winning threshold are “wasted” because they don’t help the party win the seat. Similarly, every vote cast by the losing party is also “wasted.” To create a partisan gerrymander, a party seeks to distribute its votes efficiently and force its opponent to waste large numbers of votes. The difference, or gap, in votes wasted by each party can then be compared, with large gaps indicating possible gerrymanders.

Related: "Slaying the Partisan Gerrymander"

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