An Objective Judicial Standard for Partisan Districting or How to Tell a Gerrymander When We See One

James Allison
Professor Emeritus, Psychology, Indiana University-Bloomington

In the 2018 election 56% of Indiana voters won 78% of the state’s 9 seats in the U.S. House of Representatives. Such a lopsided coup seems plain unfair and a clear violation of equal representation, where every vote should weigh the same as any other. My question here is whether such disproportionality is a significant departure from the standard of equal representation, or mere run-of-the-mill bad luck, like five heads in a row from a perfectly fair coin.

For an objective answer I used election data from the Indiana Secretary of State, starting with records dated January 1, 2019, for the 9 Congressional races. From each of the 9 districts I took the number of votes cast for each major-party candidate, and calculated the number of Congressional seats won per vote. If all votes weighed the same, this seats/vote metric would have the same value for all voters. On the contrary, I found that voters of Party X won significantly more seats per vote than those of Party Y. The results are in the table below. (Because seats/vote is an inconveniently small number I multiplied each one by 100,000 or, for other offices, some suitably smaller number.)

(Congressional Seats/Vote)(100,000)

District X Y
1 .00 .63
2 .80 .00
3 .63 .00
4 .64 .00
5 .56 .00
6 .65 .00
7 .00 .71
8 .64 .00
9 .65 .00
N 9 9
Mean .51 .15
t = 2.58, p < .05

The bottom line of the table means that a party difference as big and consistent as this one should not be dismissed as a long run of heads from a perfectly fair coin. The probability of such a difference is simply too small, p < .05. More likely, the two party distributions represent a true consequential difference, not a negligible one.

In scientific circles a p value greater than .05 would usually open the door to the possibility of a negligible difference, one attributable to mere chance. In the present case, that door is closed. In Indiana’s 2018 election for Congress, a voter of Party X had significantly more power than one of Party Y. That is not what Americans mean by representative government. The party difference raises the distinct possibility that in our most recent redistricting, in 2011, the legislature responsible for the new district map drew it up with a deliberate eye toward partisan advantage, a practice widely called “gerrymandering.”

In past legal challenges to such maps, courts have sometimes balked at the lack of a judicial standard for partisan redistricting. In effect, a court may ask, “How can we recognize a gerrymander when we see one?” How big a departure from equal representation must we endure before a court cries “enough” and orders a more equitable map? It seems to me that we already have a practical judicial standard, firmly based on the mathematical logic of inferential statistics applied to a simple metric, seats per vote.

We can probe the validity of that metric by tracking its response to a more equitable district map of that same Congressional election. As it turns out, by changing a mere 3% of the vote distribution, only 59,000 votes, we could render the number of seats a party would have won strictly proportional to its popular vote. The result would be a Congressional delegation more representative of its constituents.

Specifically, suppose we change the election returns of Party X by moving 23,000 of its votes from District 2 over to District 3, and 36,000 from District 9 to District 8. The popular vote totals of the two parties remain the same as before, but their Congressional representations become exactly proportional to the popular vote: 56% of the seats for Party X, and 44% for Party Y. And seats/vote, our proposed judicial standard, which previously differed significantly between parties, is now statistically equal between parties. The results appear below.

(Congressional Seats/Vote)(100,000)

District X Y
1 .00 .63
2 .00 .97
3 .55 .00
4 .64 .00
5 .56 .00
6 .65 .00
7 .00 .71
8 .52 .00
9 .00 .85
N 9 9
Mean .32 .35
t = 0.17, p > .05

Thus, under our hypothetical new district map Party X, with 56% of the popular vote, wins 56% of the Congressional seats (5/9); Party Y, with 44% of the popular vote, wins 44% of the Congressional seats (4/9); and our proposed metric, seats/vote, reveals no significant difference between the two parties (t = 0.17, p > .05).

What does our metric show about the political composition of the Indiana Legislature following the 2018 election?

The Indiana Assembly comprises 100 members elected from 100 districts every four years. The results of the 2018 election, compressed into ten groups of the consecutively numbered districts, appear below in the form of the seats/vote metric. They look much like the Congressional results, with Party X capturing significantly more Assembly seats/vote than Party Y. Accordingly, Party X, with only 56% of the popular vote, captured 67% of the seats in the assembly, and Party Y, with 44%, only 33% of the seats.

(Assembly Seats/Vote)(100,000)

District X Y
1-10 .42 .66
11-20 .52 .40
21-30 .58 .27
31-40 .53 .29
41-50 .71 .00
51-60 .66 .00
61-70 .55 .25
71-80 .57 .37
81-90 .58 .18
91-100 .78 .58
N 10 10
Mean .59 .30
t = 3.82, p < .01

But if we shift just 2% of the votes from 12 Party X districts into 12 other Party X districts we can achieve strict proportionality, where Party X, with 56% of the popular vote, captures 56% of the seats in the Assembly (56/100), and Party Y, with 44% of the popular vote, captures 44% of those seats (44/100). That change is accompanied by a parallel change in the mean value of our seats/ vote metric: Previously, Party X voters carried significantly more weight than Party Y voters (.59 vs. .30); now they are equal, with means of .35 (Seats/Vote)(10,000) for Party X, and .35 for Party Y. Thus, by moving a few Party X votes to a few other districts we achieved proportional representation, equalized our seats/vote metric, and removed all trace of gerrymander.

That leaves the Senate races of 2018.

Half of the 50 Senators in the state Senate are elected every two years. In the 25 Senate races of 2018 Party X won 66% of the popular vote and a stunning 84% of the Senate seats (21/25). Party Y won 34% of the popular vote and a paltry 16% of the Senate seats (4/25). The results, compressed into five groups of five districts each, are shown below in terms of the seats/vote metric. Again they show Party X with significantly more seats/vote than Party Y.

(Senate Seats/Vote)(100,000)

District X Y
1, 4, 6, 11, 14 .19 .14
15, 17, 19, 21, 22 .38 .00
23, 25, 26, 27, 29 .20 .16
31, 38, 39, 41, 43 .34 .00
45, 46, 47, 48, 49 .32 .00
N 5 5
Mean .29 .06
t = 4.24, p < .01

If we shift a few votes—2% of the total Senate vote—from four Party X district totals into four other such totals we get near proportionality: Party X, with 66% of the popular vote, wins 68% of the seats (17/25) and Party Y, with 34% of the popular vote, wins 32% of the seats (8/25). In a corresponding shift, Parties X and Y become statistically equal in mean seats/vote, .21 and .12 respectively, t = 1.50, p > .05.

Should other investigations confirm these findings, it seems to me that a court or a state legislature confronted with the evidence could hardly plead lack of a judicial standard for partisan districting. The standard is there, based firmly on the mathematical logic of inferential statistics. This is how we can tell a gerrymander when we see one.