The Signature of Gerrymandering

Our Quantifying Gerrymandering group at Duke generated an ensemble of over 24,000 redistricting plans, sampled from a probability distribution placed on the collection of redistricting plans.  The ensemble was used to evaluate the 2012 and 2016 congressional district plans enacted by the NC General Assembly.  The two enacted plans were both found to be statistical outliers in the context of the ensemble of 24,000 plans; this outlier analysis formed the central argument of Jonathan Mattingly’s testimony in Common Cause v. Rucho.

In the outlier analysis, the most obvious statistic to consider is the partisan makeup of the congressional delegation each map produces.  The following histograms show that the 2012 maps (NC2012) and 2016 maps (NC2016) produce unlikely results.  In contrast, a map produced by a bipartisan panel of retired judges (Judges) produces typical results.

However, this simple analysis does not tell a complete story: In particular, as shown in the discussion of Firewalls,  a map can produce quite typical results for some elections and outlier results for other elections.

When analyzing the ensembles of predicted election results, different elections probe different elements of a redistricting plan’s structure.  A redistricting plan yields atypical election results only when the plan’s overall structure is anomalous in a way that is relevant to a particular election.   In short, the same plan can yield both anomalous and typical results for different elections, however some plans always give typical, expected results.

To make this conversation concrete,  let us consider the summary graphic which was central to our court testimony and identifies the signature of gerrymandering — namely the box-plot of ordered marginal distributions.

The construction of the marginal box-plots is fully discussed in a separate blog post. It is a visual way to summarize the typical distribution of the partisan make-up of a districting plan, and allows us to recognize maps with anonymous structures.  We can further use the box-plots to understand when a given map will produce atypical election results.

Box-plot summary of districts ordered from most Republican to most Democratic, for the voting data from 2012 (left) and 2016 (right). We compare our statistical results with the three redistricting plans of interest. The values for each map have been connected with  lines to emphasis their structure.

To interpret maps, one needs mainly to understand that the “boxes” on the box plot contain 50% of the ensemble, while the  extent of the whiskers give the max and min with the inner ticks marking  90% and 10% quantiles and the outer ticks marking the 97.4% and 2.5% quantiles.  Hence maps with most of their dots inside the boxed regions are seen as typical, while plans with values far outside of the boxes are outliers.

The boxes also predict whether a district would have typically elected a Democrat or Republican: when a box falls below the 50% dashed line (marked 0.5 Democratic vote fraction), that district will typically elect a Republican, where as when the box falls above the 50% line, that district will typically elect a Democrat.  Boxes falling over the line suggest a district that may elect either party.  We can similarly examine a given plan in the same way, except we have a point rather than a box, and the result will be fixed because it will either be above or below the 50% line.

Because the districts are ordered the box-plots predict how many officials from each party will be elected.  For example, in the 2012 votes, we expect between 6 and 7 democrats to be elected; when comparing the results of the ensemble with the NC2012 map, we see there are only 4 Democrats elected.

The box-plots reveal much more than the number of elected Democrats/Republicans.  They also reveal deep structural differences between plans.  The atypical sigmoidal curves formed by the NC2012 and NC2016 plans reveal a deep structural difference from the typical plan in the ensemble.  This can be seen clearly by examining the yellow curve obtained by connecting the medians of each box.  The atypical curves were dubbed  the “shape of Gerrymandering” by Common Cause lawyer Steve Epstein, and, as we will demonstrate below, have significant consequences for the responsiveness of elections.


Non-responsiveness to changes in the Electorate

When public opinion sways to a political party, the statewide vote fraction for that party will increase.  When estimating how changes to the statewide partisan vote fraction will effect the district level partisan vote fractions, one may think about simply shifting the results in each district by a proportional number of percentage points.  Under this assumption, a 2% swing to the Republicans would increase the Republican vote fraction in each district by 2% and decrease the Democratic percentage by 2%. This amounts to sifting each box-plot down by 2%. If the swing had been for the Democratic Party it would have shifted the box-plot up by 2%.

Given the above assumption along with the large jump between the tenth and eleventh most democratic districts observed in the NC2012 and NC2016 district plans, there will be a large range of elections (with wildly different state-wide partisan vote fractions) that produce a Congressional delegation with the same partisan make-up — in this case, 3 Democrats and 10 Republicans.


In contrast, the redistricting plan proposed by the Judges has no large gaps. It gradually increases gradually from the left to the right. As a consequence, shifts in the global partisan sentiment lead to Congressional seats gradually shift from one party to the other.  The Judges plan has a curve which is very close to the plot connecting the medians, meaning that for all statewide partisan vote fractions, the split between partisan seats will be reflective of the ensemble.

Packing and cracking

The box-plots also allow us to identify districts which have had one party packed in or diluted. Using the boxplots in this way is discussed more in the post of box-plots.