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Three tests for practical evaluation of partisan gerrymandering.

3. National districting patterns can be used to identify a natural seats/votes relationship

Computer simulations can be used to ask a simple question: If a given state's popular House vote were split into differently composed districts carved from the same statewide voting population, what would its congressional delegation look like? The answer allows the definition of a range of seat outcomes that would arise naturally from districting standards that are extant at the time of the election in question.

It is possible to calculate each state's appropriate seat breakdown--in other words, how a congressional delegation would be constituted if its districts were not contorted to protect a political party or an incumbent. This is done by randomly selecting combinations of districts from around the United States that add up to the same statewide vote total for each party. Like a fantasy baseball team, a delegation put together this way is not constrained by the limits of geography. On a computer, it is possible to create millions of such unbiased delegations in short order. In this way, one can ask what would happen if a state had districts whose distribution of voting populations was typical of the pattern found in rest of the nation. (108) Because this approach uses existing districts, it uses as a baseline the asymmetries that are present nationwide. (109) Indeed, the average result of these simulations approximates a "natural" seats/votes relationship that can be defined with mathematical rigor and exactitude. In short, these simulations detect distortions in representativeness in one state, relative to the rest of the nation.

Using a standard ThinkPad Xl Carbon laptop computer equipped with the mathematical program MATLAB, simulation code (110) can perform one million simulations for a state in less than twenty seconds. Figure 2 shows one thousand such "simulated delegations" for the state of Pennsylvania, along with the actual outcome in gray. The solid curve defines a mathematically expected average seats/votes relationship.


It is apparent that most possible redistrictings would have resulted in a more equitable congressional delegation. For outcomes with the same popular-vote split (50.7% Democratic, 49.3% Republican), one million simulations gave a median result of eight Democratic, ten Republican seats (an average of 8.5 Democratic seats). The actual outcome was five Democratic, thirteen Republican; however, only 0.2% of the simulations with the same popular vote (i.e., 50.7% Democratic) led to such a lopsided (or a more lopsided) split favoring Republicans.

Pennsylvania is known to have been targeted by the Republican State Legislative Committee's Redistricting Majority Project (REDMAP), a multiyear effort to facilitate and carry out aggressive redistricting after the 2010 census. (111) A similar computational analysis of all fifty states can be done to test if additional REDMAP states show statistical anomalies.

For all fifty states, Figure 3 is calculated using the vote outcomes of non-extreme states (shaded in light gray) to feed the simulations. (112) These results coincide strongly with targeted partisan redistricting efforts (113) and are highly unlikely to have arisen by chance. White shading indicates Republican Party control over redistricting, dark gray indicates Democratic Party control, and black indicates nonpartisan commission (AZ, Arizona) or a court-ordered map (TX, Texas). Out of ten states with extreme outcomes, eight favored the party that controlled the process, and none worked against the party in control. (114) Indeed, the extreme cases include all states with single-party control that have been mentioned on a redistricting watchlist published in 2011 by the Washington Post. (115)


In Part II.B below, I develop an analysis of intent that again uses the zone-of-chance concept. There, as here, the standard deviation, sigma (a), will be used as a yardstick of deviations from the average expected outcome. As before, the general idea is that an average outcome only reflects one point in a range of outcomes, and the standard deviation is necessary to define a zone of chance. Generally speaking, for a bell-like curve, which these simulations approximately follow, a difference of 1.6 standard deviations or more occurs by chance in five percent of cases. Five percent is a common threshold for determining statistical significance. (116) The standard deviation is a handy and universal reference measure for detecting extreme outcomes, and it applies to all the analyses and tests in this Article. For convenience of notation in the tables that follow, I define the quantity Delta ([DELTA]) as the difference from average expectations, divided by sigma.

Table 1 shows states for which the partisan discrepancy was greater than one sigma in 2012. For comparison, discrepancies for the same states are shown for 2010 and 2014. Simulation-based values for sigma are given in the columns labeled "SD (sigma)." (117)

Five states showed deviations that were greater than one sigma and less than two sigma: Florida, Illinois, Indiana, Maryland, and Virginia. Six more states showed a deviation exceeding two sigma: Arizona, Michigan, North Carolina, Ohio, Pennsylvania, and Texas. Of these eleven states, REDMAP's redistricting efforts are known to have targeted five: Indiana and all four Republican-controlled states with two-sigma discrepancies, namely Michigan, North Carolina, Ohio, and Pennsylvania. (118) Of the remaining greater-than-two-sigma states, a fifth state, Texas, was redistricted by Republicans but showed a discrepancy favoring Democrats. (119) A sixth state, Arizona, was redistricted by an independent commission and favored Democrats. (120)

Of these six states, I briefly describe three cases of special interest: California, Texas, and Florida.

California. California is worth mentioning as a counterexample to the imbalanced states shown above. California was redistricted by an independent commission. (121) In 2012, the California House popular vote was 62% Democratic, resulting in 38 out of 53, or 72%, Democratic seats. (122) However, the average simulated delegation was also 72% Democratic. (123) Thus, election results in California exactly meet the expectations that arise from nationwide districting patterns.

Texas. Although the resampling simulations are a powerful and sensitive measure, the case of Texas demonstrates how examination of additional factors can be necessary. Before the 2012 election in Texas, a complex series of legal battles culminated in a court-ordered redistricting plan (124) and a congressional election outcome in which over 60% of Texas voters voted for Republicans, resulting in 24 Republican seats out of 36 total. (125) From a statistical standpoint, this was an underperformance for Republicans, who in a simulation would have won over 28 seats on average--a discrepancy of Delta = 2.3 times sigma, which is outside the zone of chance, and therefore a statistically significant deviation. One major factor contributing to this discrepancy was the presence of Hispanic majorities in 9 districts, (126) 6 of which elected Democratic congressmen. (127) These majority-minority districts, which have special status under the Voting Rights Act of 1965, reflect the growing Hispanic population in Texas, which as of the 2010 census constituted 38% of Texans. (128) Democrats won approximately 40% of the statewide two-party popular vote and won 12 out of 36 seats, or 33% of seats. (129) Because this change is in the direction of proportionality compared with typically occurring seats-votes curves, it is eu-proportional. The number of majority-minority districts (which usually favor Democrats) falls within the Gingles criteria. Thus, the final outcome in Texas in 2012 favored the partisan minority for mandated race-based reasons, and because it is eu-proportional, would not be grounds for further action.

Florida. In this case, where the value of Delta is between one and two, a similar but statistically stronger answer is given by a map-drawing approach. Chen and Rodden took a geographically intensive approach, drawing districts using automated rules of contiguity and community preservation, and implemented these rules thousands of times through detailed computer simulations. (130) They found that Florida's 2010 redistricting scheme was more favorable to Republicans than over 99% of their simulations, indicating that the Florida legislature applied an approach that led to a more partisan outcome than Chen and Rodden's rules would support. (131) Geographic considerations are among the principles of districting mandated by the Constitution of the State of Florida, which also allows for judicial review by the Florida Supreme Court. (132) In July 2015, that court replaced the map to comply with the state constitution. (133)

Nationwide, repairing the one-sigma and greater Republican-redistricted states (seven in all) would lead to an average swing of approximately twenty-eight seats (an average of 27.7) toward Democrats; repairing the two Democratic-redistricted states, Illinois and Maryland, would lead to an average swing of 5.7 seats toward Republicans. Therefore, based on these measures, Republican gains in 2012 from aggressive redistricting (28 seats) were nearly five times the advantages gained by Democrats from the same process (6 seats). This sharp asymmetry coincides with a period during which state legislative processes have come increasingly under single-party control. (134) Changes between decadal redistrictings favored Republicans, who controlled 13 state capitals in 2002, rising to 24 state capitals in 2012. (135) During that same interval, Democrats went from controlling 8 state capitals to controlling 13 state capitals. (136) Thus the potential for partisan control of districting has increased for both major parties, with a greater advantage for the Republican Party.

4. What accounted for the antimajoritarian outcome of 2012?

With these analytical tools in hand, it is now possible to calculate the total effect of asymmetric partisan districting on the national House elections of 2012. The outcome was a 33-seat margin of control, with 234 Republican, and 200 Democratic seats. (137) Applying party-neutral standards to the 7 Republican-controlled states and 2 Democratic-controlled states would have given an average margin that was 22 seats smaller, or 212 Democrats and 223 Republicans. Because of the uncertainty contained in this analysis (the range of outcomes within two sigma of the average was 206 to 218 Democratic seats), it is just within the range of possibility that without partisan asymmetry, Democrats might have taken control of the chamber.

Republicans have a second advantage, one that arises from population clustering. Because this simulation-based analysis uses existing national districts, it includes effects of population clustering. It is possible to quantify the net impact of population clustering, which facilitates the drawing of districts that are heavily tilted toward Democrats. (138)

In the original simulations, states where I did not find dys-proportionality had a two-party vote share of 50.7% for Democrats and 180 out of 363 seats. I then calculated the expected share of seats if district-by-district vote shares were perfectly symmetrically distributed. (139) Such symmetry of population patterns predicts that a 50.7% vote share would lead to Democrats winning 51.8% of seats, or 188 seats, 8 seats more than the real-population-based simulation. Scaling this up to all 435 seats, this suggests that Republicans won 9 or 10 seats in non-dys-proportional states more than they would under symmetric population patterns; the swing, defined as margin between the parties, is twice as large--18 to 20 seats. This 18-to-20 seat swing effect across all fifty states is smaller than the 22-seat effect of partisan dys-proportionality in just nine states. (140) Therefore although a considerable deviation from natural seats/votes relationships is driven by political geography, an even larger total effect arises from political motivations and actors during the legislative process in just a handful of states.

In summary, the effects of partisan redistricting exceeded the amount of asymmetry caused by natural patterns of population. Together, gerrymandering and population clustering are more than enough to account for the fact that in 2012, Democrats won the House popular vote but Republicans ended up in control of the chamber.

B. Analysis of Intents: Voter Packing by Intentional Gerrymandering and Self-Association

The Analysis of Effects (discussed in Part II.A above) established a method for identifying states in which voter preferences lead to representation that is anomalous relative to national norms. Without gerrymandering, these anomalies could be rectified through the ballot box: if election outcomes shift sufficiently, legislators can be voted out, thus bringing outcomes more in line with the popular will. As an example of how gerrymandering vitiates this mechanism, the election of 2014 heralded a "wave year" in which Republicans won the national popular vote by 5.9%, in sharp contrast to the Democratic popular vote win of 2012. (141) However, in the twelve states in Table 1, Republicans gained control of only 5 of 187 Democratic seats. (142) This small change indicates that representatives in these states were largely insulated from a large swing in opinion from 2012 to 2014. Considering the strength of partisan gerrymandering in 2012, the smallness of this change means that Republicans reaped most of their electoral gains two years earlier than their popular support would have merited.

The Analysis of Intents below presents a way to identify asymmetric reductions in the ability of legislative elections to respond to changes in voter opinion. It therefore can be used to measure a principal effect of partisan gerrymandering: reduction in the overall responsiveness of races across a state. I propose that such a pattern is indicative of the intentions of the entity that drew the district lines.

1. Distinctive patterns of win and loss margins arising from partisan gerrymandering and voter self-association

Partisan redistricting procedures create a characteristic lopsided pattern of election results that can be used to identify when packing is likely to have occurred. State-level gerrymandering is more elaborate than single-district gerrymandering and relies on a two-part strategy. First, as before, map-drawers will cram "voters likely to favor [their] opponents into a few throwaway districts where the other side will win lopsided victories, a strategy known as 'packing.'" (143) Second, map-drawers will draw the remaining, more numerous districts using boundaries to yield more narrowly won victories. (144) In this process, the critical requirement is asymmetry: the opposing party's voters must be more tightly packed than one's own voters. (145) The net result is an increased likelihood of unrepresentative outcomes.

I examine lopsided patterns in gerrymandered states and compare them to nongerrymandered states: this provides a comparison of the effects of partisan gerrymandering with the effects of population variations and less-partisan districting. This analysis will be used as the basis for Test 2 in Part III.A, an index of gerrymandering that depends directly on the partisan redistricter's desired goal: the packing of opponents, as measured by election returns.

Gerrymandered districts show a distinctive pattern of lopsided votes (Figure 4). Figure 4a shows a histogram of two-party vote share for 2012 House districts that were asymmetric in favor of Republicans. In this histogram, two peaks are apparent: a narrow peak centered near a 40% Democratic vote share and a broader peak centered near a 30% Republican vote share (indicated on the histograms by a 60% to 80% Democratic vote share). Both of these peaks are sufficiently prominent that they can also be seen in a histogram drawn using all states nationwide (Figure 4a). The peaks are considerably more prominent when the histogram includes only Republican-favoring states (Figures 4b and 4c) or Democrat-favoring states (Figure 4d).


However, as stated in Part II.A.4 above, voter packing can be asymmetric simply by virtue of the fact that voters arrange themselves in a manner that is not symmetric. Therefore any measure of gerrymandering-based packing must be done relative to a baseline of how voters "pack themselves." Specifically, it has been suggested that structural factors such as concentration of Democrats in urban areas may have a greater effect than partisan redistricting. (146) I now show how these two effects are manifested in district-level outcomes. Since both real packing by redistricters and virtual packing by structural factors are likely to have similar manifestations, they can be examined using the same statistical tools.

2. Gerrymandering emulates and amplifies the representational consequences of urbanization

The establishment of competitive districts is made difficult by the fact that voters often choose to live near others of similar ethnic, religious, secular, and political affiliation. (147) Such self-selection is visible in urban regions that vote overwhelmingly for Democrats and rural regions that vote overwhelmingly for Republicans. If natural population clustering favors increased Republican representation, then the distribution of vote share in urbanized districts should resemble that of Republican-gerrymandered states. Such a pattern is not apparent in high-population-density states (Figure 4e). However, urbanized districts (Figure 4f), defined as those with population density greater than 1000 persons/square mile, show both peaks, but with more emphasis on the high-Democratic-vote share peak. This pattern is visible even when putatively gerrymandered states (favoring both Democrats and Republicans) are omitted from the histogram (Figure 4g).

Gerrymandering makes use of existing urbanization. In Republican-gerrymandered states, non-urbanized districts (Figure 4b and 4c) are dominated by Republican-packed districts, demonstrating that redistricters who seek a Republican advantage do so by creating numerous districts that avoid urban regions. Once Republican gerrymanders and urbanized areas are omitted, a histogram of the remaining congressional districts no longer has two peaks (Figure 4h). Democratic gerrymanders can achieve a converse advantage by carving out slices of urbanized areas and combining them artfully with more rural areas to create small but secure Democratic wins.

Although the representational effects of voter migration into urban communities are similar to the effects of partisan gerrymandering, the interpretations of the two phenomena are quite different. Voters who arrange themselves in this manner are voluntarily arranging themselves so that their representatives are at little risk of being turned out of office. In the case of partisan gerrymandering, voters are placed into political affiliation with one another--but without the consent of the citizens involved. Such a pattern contradicts the saying that "voters should choose their representatives, [and] not the other way around." (148) Gerrymandering thus penalizes voters based on their publicly available information, including partisan loyalty, which is present in census data and commercial redistricting software.

3. A "lopsided-margins test" to detect when the targeted party wins with unusually large margins

In summary, the success of a gerrymandering scheme depends on the ability of the redistricting party to create safe margins of victory for both parties, with larger margins for their opponents. This pattern of outcomes can be quantified by sorting the districts into two groups, by winning party. Each party's winning vote shares can then be compared by what is said to be "the most widely used statistical test of all time" (149): the f-test for comparing the averages of two groups of observations. In this way, the difference between each party's winning margins will be used to carry out Test 2, which tests for intensive packing of one party's voters.

4. The mean-median difference as a measure of skewness

Now that I have identified states in which Republicans or Democrats gained an asymmetric advantage, I can examine these states to test the validity of a simpler statistic that does not require computer simulation: the difference between the mean (i.e., average) and the median vote share (150) for contested districts. (151) The mean-median difference is a simple measure of asymmetry (152) that allows for ready comparison with national standards. Notably, it does not require any inputs other than district-level election results for the state that is under examination.

As an example of the calculation, consider the 2012 Pennsylvania congressional election. The Democratic two-party share of the total vote in all eighteen districts was, in terms of percentages and sorted in ascending order:
   34.4, 36.0, 36.6, 37.1, 38.3, 40.6, 41.5, 41.6, 42.8, 42.9, 43.1,
   43.4, 45.1, 48.3, 60.5, 69.1, 76.9, 84.9,90.5. (153)

Races won by Republicans are indicated in italics and the two middle values are underlined. The median percentage is defined as the midpoint of the two middle values, 42.85%. The mean Democratic vote share is 50.5%. The difference between the median and the mean is 7.6%. This difference reflects the fact that counterintuitively, Republican vote shares were above average in considerably more than half of the districts: 72% (thirteen out of eighteen), to be exact.

The median serves as a measure of the overall behavior of the eighteen district-level elections. The goal of a gerrymander is to maximize the number of districts won, which occurs when the median outcome is more unfavorable to the opposing party than that party's share of the vote. In other words, Pennsylvania's Democratic voters were empowered as if they comprised 42.85% of voters, even though they actually comprised 50.5%. The difference, 7.6%, is the number of voters who were effectively disenfranchised. Since approximately 5.5 million Pennsylvanians cast votes in the 2012 congressional election, redistricting achieved an effect equivalent to over 400,000 Democratic voters casting their ballots for Republicans. The probability is less than 1% that this difference arose by chance. (154)

5. State-by-state comparisons of skewness with population clustering effects

To investigate the degree to which the mean-median difference arises as a function of population clustering patterns, I make comparisons between a variety of states and years. For the 2012 congressional elections, the nationwide mean-median difference was 4.3% favoring Republicans across all fifty states and 1.9% favoring Republicans in non-dys-proportional states. For Pennsylvania in 2012 the difference was 7.6%, greater than any of the other mean-median differences, and comparable to the other four dys-proportional states of Michigan (mean-median difference of 6.3%), North Carolina (7.3%), Ohio (155) (6.3%), and Virginia (6.3%). Overall, these mean-median differences are three to four times those seen in non-dys-proportional states, indicating that within a single state, the effects of partisan gerrymandering can be three or four times larger than the effects of population clustering. Indeed, as stated previously, redistricting in a handful of states can generate a greater deviation from symmetry than population clustering in all fifty states combined.

III. Three Quantitative Tests for the Detection of the Effects of Partisan Gerrymandering

I use the two Analyses to propose three tests. The three tests have several advantages. First and foremost, they are simple to apply. None of the three tests requires the detailed drawing of maps. Because the tests can be stated with mathematical exactness, they can also provide a manageable standard for gerrymandering cases, yielding predictable and sensible results--and unambiguous guidance to legislatures and judges. The tests are based on election outcomes and therefore can be employed separately from, or in conjunction with, geographic and other criteria. An online calculator for these tests is available at

A. Converting the Analyses to Practical Tests

I use the Analysis of Effects, which is based on numerical simulation of seat outcomes, to construct Test 1, the excess seats test. 1 use the Analysis of Intents, which identifies narrow-but-reliable wins as a hallmark of gerrymandering, to construct two tests: Test 2, the lopsided outcomes test; and Test 3, a reliable-wins test.

Test 1 (the excess seats test): Calculate whether the outcome of an election after redistricting was dys-proportional relative to a simulated seats/votes curve and whether that outcome favors the redistricting party. For a state containing Ndistricts, calculate the difference between the actual seats and the simulated expected number and divide by the standard deviation to obtain the difference, Delta. (156)

Test 2 (the lopsided outcomes test): Compare the difference between the share of Democratic votes in the districts that Democrats win, and the share of Republican votes in the districts that Republicans win. This test works because in a partisan gerrymander, the targeted party wins lopsided victories in a small number of districts, while the gerrymandering party's wins are engineered to be relatively narrow. (157) To compare the winning vote shares for the two parties, I use a grouped f-test, an extremely common statistical test. (158)

Test 3 (the reliable-wins test): Systematic rigging of total statewide outcomes occurs by the construction of districts that offer secure wins for the party in control of the map. These wins would be wide enough to guarantee victory, but not so wide as to waste votes that could be used to shore up other districts. How this intent is detected depends on whether the state's partisan vote is closely divided, or whether one party is dominant. In a closely divided state, reliable wins occur when the average and median vote differ from one another. In a state that is dominated by one party, reliable wins occur when that party's strength is spread highly evenly across districts.

* In a closely divided state-. Calculate the difference between a party's statewide average (i.e., mean) district vote share on the one hand, and the median vote share it receives on the other. In this situation, a systematic gerrymander can be detected when a party's median vote share is substantially below its average vote share across districts. (159) For this test, calculate Delta by dividing the mean-median difference by [[sigma].sub.3], which is defined as 0.756 * (standard deviation of vote share across all N congressional districts in a state)/[square root of N]. (160)

* In a state where the redistricting party is dominant: Calculate the standard deviation of the redistricting party's vote share in the districts that it wins. Calculate the standard deviation of the party's vote share in the districts that it wins nationwide. Compare these two standard deviations using a well-established testing tool, the chi-square test for comparison of variances, (161) to define zones of chance.

Test 1 evaluates whether a party gained a significant advantage in terms of seats and calculates the size of the effect. Tests 2 and 3 determine whether the pattern of data could have arisen by chance; if not, this indicates an intent to gerrymander. A residual possibility exists of a false-positive result, i.e., identifying that a gerrymandering event occurred when in fact it did not. To reduce the possibility of such a false alarm, partisan gerrymandering could be assessed by evaluating both Test 2 and Test 3. If Delta is set to standard levels of statistical significance (162) in 2012, six states met both the Test 2 and Test 3 criteria: Michigan, North Carolina, Ohio, Pennsylvania, Virginia, and Wisconsin.

B. Advantages and Disadvantages of the Three Tests

The tests proposed here all have several advantages. First, the tests do not require the detailed drawing of maps. Second, because they are derived from election results only, the tests can be applied independently of evaluation of intent. Third, because their results are highly correlated, in situations where one test is unsuitable, another can be used instead. In this way the tests can be used separately or combined to reduce the risk of falsely identifying a gerrymander where none occurred, or conversely, failing to detect a gerrymander where one did occur. Finally, because the three tests do not use geography, they can easily be combined with other standards which may require circuitous geographic boundaries, such as state-mandated requirements, (163) section 2 of the Voting Rights Act, and other precedents that exist in federal law.

In choosing which test to apply, the judge (or other evaluator of a districting plan) should take the following advantages and disadvantages into account.

Test 1's most powerful use is to obtain an exact range for the appropriate number of seats for a given vote share. It addresses whether a redistricting scheme leads to an elected delegation that deviates from national districting norms; i.e., it measures effects. Test 1 can always be calculated from a set of election returns. Because it uses data from other states, it has the advantage of taking into account the overall nationwide demographic character of districts. Therefore, it has the virtue of measuring effects that are in addition to those that arise naturally from population clustering. However, because it requires computer simulation, it requires the use of a computer program (which is freely accessible at and can also be obtained by contacting the Author).

Test 2 has the advantage of simplicity: it can be worked out using a spreadsheet program such as Microsoft Excel that can perform a two-sample t-test. If such a program is not available, it can be done using a hand calculator and a table of statistical values. It directly tests for noncompetitive races, a mainstay of gerrymandering. It identifies partisan asymmetry, though not bipartisan gerrymanders in which individual candidates of both parties benefit. Test 2 has the disadvantage that it can only be used if both parties win at least 2 seats each, since this is required to calculate standard deviations, a necessary step of the test.

Test 3 measures the reliability of wins for the redistricting party. Like Test 2, it is simple to calculate. Test 3 can always be done, since it is calculated using a state's district-level results. In the case of the mean-median difference, it does not rely on any data from other states and is therefore self-contained. In the case of the chi-square test, national data must be used to provide a standard for comparison.

C. Three Examples: The Original Gerry-mander, Arizona State Legislative Districts, and Maryland Congressional Districts

To examine the general applicability of these tests, let us consider three examples: (l) the original Gerry-mander of 1812, (2) the post-2010 Arizona state legislative districts that was considered by the Supreme Court in the 2015 Term, (164) and (3) the post-2010 Maryland congressional districts, which the Supreme Court recently remanded for consideration by a three-judge court. (165)

Example 1: The original "Gerry-mander," the Massachusetts State Senate Election of 1812. Test 1 is evaluated by starting from the fact that there were 18 races. (166) The average expectation of a nearly evenly divided popular vote is 9 seats for each party. The upper theoretical value to sigma is 0.5* [square root of 18] = 2.1 seats; computational simulation reveals a true value of sigma of 1.4 seats. The Federalists won only 5 seats, (167) and therefore Test 1 is met to a standard of (9-5)/1.4 = 2.9 sigma, statistically significant.

For Test 2, the Federalists won five races (which accounted for eleven districts); in these races, their two-party vote share averaged 55.6%, with a standard deviation of 4.6%. The Democratic-Republicans won thirteen races (which accounted for twenty-nine districts), with an average vote share of 70.7% and a standard deviation of 5.3%. The resulting Delta (also called t-score) is 5.5, and therefore Test 2 is met to a standard of 5.5 sigma. This is an unusually high level of significance; this result is reached by chance 0.0025% of the time.

Test 3 should not be used because districts are not equal in size. In 1812, the number of votes per legislator ranged from Dukes/Nantucket (1078 votes cast in total for one legislator) to Franklin (4469 votes for one legislator). (168)

Example 2: Arizona State Legislative Districts. After the 2010 census, the Arizona Independent Redistricting Commission, which is composed of members of both major political parties, drew House and state legislative districts. (169) A case recently decided before the Supreme Court, Harris v. Arizona Independent Redistricting Commission, concerned whether "the desire to gain partisan advantage for one political party justifies] creating legislative districts of unequal population that deviate from the one-person, one-vote principle of the Equal Protection Clause." (170) In that case, plaintiffs contended, and the District Court of Arizona "assume[d] without deciding[,] that partisanship is not a legitimate reason to deviate from population equality." (171) The Independent Redistricting Commission, by contrast, contended that it constructed districts of unequal population to comply with section 5 of the Voting Rights Act. (172) In a unanimous opinion, the Court held for the Commission, allowing the district map to stand. (173)

Although the issue at hand was the creation of overpopulated districts, neither side contested in federal courts the premise that the Commission created a partisan advantage. This turned out to be a key point, since the Court noted that "[e]ven assuming, without deciding, that partisanship is an illegitimate redistricting factor, appellants have not carried their burden." (174) The question bears examination: Did redistricting actually create a partisan advantage in the first place? This question can be tested by examining state senate races, of which there is one for each of Arizona's thirty legislative districts; or state House races, which elected two Representatives for each of the same thirty districts.

Test 1 relies on computer simulation using other comparable districts as a source of hypothetical districts. The statewide two-party popular vote totaled 56.3% for Republicans and 43.7% for Democrats, yielding seventeen seats for Republicans and thirteen seats for Democrats. Because other states have different districting systems (for instance with different numbers of people per district), data is not available to allow simulation of the seats/votes relationship. However, a simpler calculation is possible: proportional representation would predict 16.9 seats for Republicans. Therefore, the election result is almost perfectly eu-proportional, and therefore does not require further analysis.

For Test 2, appellants asserted that the Democratic Party benefited from the redistricting. (175) In Arizona's state senate races in 2014, the average winning Republican vote share was 73%, while the average winning Democratic vote share was 72%. (176) This difference--one percentage point--is not statistically significant. In state House races, Republicans won with an average of 66% in those districts they won, while Democrats won with 64% in those districts they won; again, the difference was not statistically significant.

For Test 3, the mean Democratic vote share across thirty districts was 50.1%. Therefore, the state's votes were closely divided, and the appropriate test is the mean-median difference. The median Democratic vote share was 45.6%, for a mean-median difference of 3.3% (4.1% with imputation of uncontested races) in a direction that favors Republicans. This difference works against Democrats and therefore is in the wrong direction.

Based on the foregoing, Arizona senate districts fail all three tests. Therefore, the contention that Democrats benefited in a dys-proportional manner is not supported, and the Supreme Court was correct in pointing out the absence of undue partisanship. If the Commission was trying to engineer a map that systematically disfavored Arizona Republicans, it did a poor job.

Example 3: Maryland Congressional Districts. Maryland has eight congressional districts. Steven Shapiro and other plaintiffs filed suit in district court that the post-2010 districting plan violated their rights to political association under the First Amendment. (177) The district court dismissed the complaint, and the U.S. Court of Appeals for the Fourth Circuit affirmed its dismissal. (178) However, in December 2015 the Supreme Court reversed the decision, remanding the case to a three-judge court for further consideration. (179)

In Maryland, Democrats typically win around 60% of the vote at a statewide level--the same as the margin needed for a safe victory. Artful arrangement is accomplished and can be detected in the form of many districts of near-identical partisan composition (Figure 5).


Test 1 identifies Maryland as a gerrymander. In the pre-redistricting election of 2010, Democrats won 63.2% of the statewide vote and six seats, (180) compared with a simulated average of 6.1 seats--not statistically significant (Table 1). However, after redistricting, in 2012 Democrats won 65.5% of the statewide vote and won seven seats, (181) compared with a simulated average of 6.1 seats. The value of Delta was 1.2 favoring Democrats, not quite statistically significant. In 2014, Democrats' vote share declined to 58.1%, but they retained all seven of their seats. (182) In this case, the simulated average was 5.1 seats, and the value of Delta was 2.4, statistically significant. These results indicate that redistricting gained Democrats a one-seat advantage in 2012, a strong Democratic year, and that this advantage was retained in the national wave election of 2014 that swept dozens of Republicans into office in states outside Maryland.

Test 2 cannot be applied because the standard deviation of the Republican winning vote share cannot be calculated with only one Republican congressman; at least two values are required to calculate a standard deviation.

Test 3 should be done for the case of partisan dominance, a situation that calls for the chi-square test to test whether Democratic votes are spread unusually uniformly across congressional districts. Figure 6 shows the standard deviation. (183) The standard deviation of Maryland Democrats' winning vote share in seven districts was 6.6% in 2012 and 7.3% in 2014. I compared the variability of Maryland Democratic districts with the variability of Democratic districts nationwide. The values for Maryland fall outside the zone of chance.


Maryland's standard deviations would have arisen by chance in only 2.8% of cases in 2012, and 1.7% of cases in 2014. (184) A third year, 2004, also showed an unusually low standard deviation. (185) These findings show that the Democrats' partisan advantage was achieved by spreading their partisan support in a highly even manner across their winning districts.

IV. Discussion

In this Article, I have presented three tests for rapid identification of partisan gerrymanders. My tests can be used to evaluate intents and effects; the two prongs articulated in Davis v. Bandemer. The two intents tests can be done with computing resources already available on a judge's or clerk's desk, and the effects test requires some additional software. (186) All three tests rely on well-established statistical principles. The tests measure different aspects of partisan asymmetry and therefore fall within the scope of principles that have been expressed by the Supreme Court. I suggest that these tests may constitute a manageable standard for courts to evaluate the impact of a state's districting scheme on its residents' equal protection and First Amendment rights.

The broader implications of this Article are threefold. First, I have used statistical science to express the idea that a pattern of election results might have arisen by chance, and therefore not warrant judicial intervention. By establishing "zones of chance" in which the partisan impacts of a districting plan are ambiguous, my tests can help a judge evaluate whether an identifiable injury has occurred in the first place. Second, my statistical analysis shows that in 2012, the effects of partisan asymmetry were so large as to exceed the effects of population clustering across the whole nation. This demonstrates the importance of measuring the degree of distortion from the natural relationship between votes and representation. Third, an intents-and-effects standard based on my tests is unambiguous and may mitigate the need to demonstrate predominant partisan intent. For these reasons, my statistical tests comprise a valuable and timely addition to the judge's toolkit for rapid and rigorous identification of partisan gerrymanders.

A. Allowing for Ambiguity

My statistical analysis of the effects of gerrymandering may be of particular relevance to First Amendment analysis, which "allows a pragmatic or functional assessment that accords some latitude to the States." (187) By allowing for a normal amount of statistical variation, the three tests proposed in this Article build in zones of chance where any of a range of outcomes would lead to an acceptable amount of asymmetry.

Any statistical approach contains some possibility of accidentally identifying gerrymandering where it does not exist ("false positives"), or missing cases where it did occur ("false negatives"). Tests may also sometimes not be usable, such as Test 2, when one party only wins one seat. For these reasons, I provide two separate tests of intents. These tests are oriented toward the outcomes of elections rather than the specifics of map boundaries or district procedures. The tests hew closely to the electoral goals of redistricters and do not rely on geographically oriented approaches which require normative assumptions of what constitutes good districting procedure.

B. What Is the Role of Intent?

Over time, the Court's decisions have set a standard for intent so stringent that it cannot be satisfied. (188) The resulting high bar to proving injury requires more than simply showing that partisanship was one of multiple factors and is a far higher bar than the evaluation of disparate impact alone. Such a demanding standard may have been appropriate in the absence of legislative guidance or a large body of court precedent. In the Bandemer and Vieth framework, the lack of simple and reliable tests made it necessary to assess the link between redistricters' actions and the injury. Indeed, current approaches to proving gerrymanders focus on intent, are diverse in approach, and sometimes do not agree with one another. (189) In contrast, a statistically based test may provide a more satisfactory route to satisfying the intent standard.

The facts in LULAC exemplify ambiguous intent. (190) For example, the Republican majority "honored" some requests by individual Democratic legislators in the districting process. (191) However, partisan gerrymandering is, by its nature, prone to satisfying such dual interests because a party as a whole has motivations that can align with those of selected individual legislators of the opposing party. (192) Therefore intent is most fairly evaluated at the state level or at the individual level, but not both at the same time. In addition, the majority in Crawford v. Marion County Election Board held that partisan intent alone is an insufficient reason to strike down voting restrictions. (193)

Quantitative measures of intent, such as my proposed Tests 2 and 3, allow the identification of patterns of districting that are highly unlikely to have arisen by chance, thereby providing concrete evidence that a legislature or other district-drawing body acted specifically to produce partisan outcomes. Satisfaction of such a rigorous standard should open the way to examining other facts as additional evidence for probable (not predominant) intent.

Furthermore, I suggest that districting can impose a burden on a group's representational rights whether or not the offense was intentional. Even where intentions are nonpartisan, bipartisan, or unknown, the effect of a districting plan with partisan asymmetry is to produce legislative blocs whose size is unrepresentative of the popular will. The construction of a reliable measure of effect provides clear guidance when an injury has taken place and a template for how the injury can be repaired. Just as a road worker may act to right an upended orange traffic cone even if she does not know how the cone came to be tipped over, a court may act when effects are sufficiently strong, as in disparate impact cases in racial discrimination cases. (194) Although partisan gerrymandering cases are governed by different doctrine (constitutional) than racial discrimination cases (statutory interpretation), both types of cases concern the issue of intent.

C. Evaluating the Partisan Impact of District Maps Before Implementation

Although in this Article I use election results to calculate the three tests, the tests could alternatively use other inputs. For example, to rule out the possibility that the tests may be influenced by variations in the quality of specific candidates, it would be possible to use district-level presidential vote shares as inputs. (195)

In current federal precedent, the need for redrawing a set of districts often relies on elections that have already taken place. However, by that time, an injury to voters has already occurred. To preempt such an injury from happening in the first place, the three tests could be calculated using information that is available before an election. Under the First Amendment rationale of not penalizing groups for their partisan preference, party registration might be used as an input to calculate the three tests. Political scientists, redistricters, and commercial redistricting software also use other variables to predict overall partisan preference; these predictions could also serve as inputs to the tests. Doing so would allow a hypothetical districting scheme to be assessed before it has passed into law.

The standards presented here can quantify the benefits of reform efforts directed at reducing the likelihood of partisan gerrymandering. One such route is the establishment of nonpartisan districting commissions that remove districting from the direct control of legislators. In 2008, California voters approved Proposition 11, which established the California Citizens Redistricting Commission. (196) The commission is composed of fourteen members who are drawn from members of the general public, including five Democrats, five Republicans, and four members who decline to state a partisan loyalty. (197) The commission's mandate is to draw districts that respect principles of contiguity, compactness, and representation of a community's interests. (198) The resulting congressional districts have become more competitive: margins of victory have become smaller, and incumbents have lost their reelection races at higher rates than before the formation of the commission. (199) Like the Arizona commission, the work of the California commission has led to closer races and more eu-proportional overall outcomes than precommission maps.

These tests could also be used in approaches that leave districting under the control of state legislators, but place constraints on how and what they produce. Such an approach has been taken in Florida, where ballot initiatives known as Amendments 5 and 6 were passed in 2010 and precleared by the Department of Justice a year later, becoming Article III, sections 20 and 21 of the Florida Constitution. (200) Together with Article III, section 16, (201) the Florida Constitution stipulates that district lines must be contiguous, compact, and use existing political geographical boundaries where available. (202) Districts also may not be drawn to "favor or disfavor a political party or incumbent." (203) The resulting plans are subject to review by the Florida Supreme Court, leading either to approval or return to the legislature for a further attempt to meet districting criteria. (204) The tests described in this Article could be useful in identifying statewide partisan favor. Individual districts would still need to be evaluated separately, for example to comply with Voting Rights Act restrictions and other principles set down in federal or state law. My tests, which address the properties of combinations of districts, can complement these other constraints without conflict.


Partisan gerrymandering distorts relationships between voting and representation that would otherwise arise naturally, generates seats that are unresponsive to shifts in public opinion, and chills the freedom of voters to associate with a political party of their choosing. The health of democratic processes would be considerably improved by reducing the ability of legislative processes to impose partisan distortions on redistricting maps. My three tests for asymmetry may contribute to a manageable standard for identifying partisan gerrymanders, with the eventual goal of reducing or eliminating them.


(2.) See The Gerry-mander, Bos. GAZETTE, Mar. 26, 1812, / treasures/images/90.6p 1 .j pg.

(3.) See Griffith, supra note 1, at 62.

(4.) Id. at 72-73; A New Nation Votes: American Election Returns 1787-1825, Tufts Digital Collections & Archives (June 24, 2009), [hereinafter Lampi Collection],

(5.) Lampi Collection, supra note 4.

(6.) Id.

(7.) Id.

(8.) Christopher Klein, A New Species of Monster, Bos. Globe (Sept. 11, 2011),

(9.) Lampi Collection, supra note 4.

(10.) Micah Altman, Traditional Districting Principles: judicial Myths vs. Reality, 22 Soc. SCI. Hist. 159,180, 187(1998).

(11.) JUSTIN LEVITT, BRENNAN CTR. FOR JUSTICE, A CITIZEN'S GUIDE TO REDISTRICTING 1,16 (2010), %20Single %20Page.pdf. For a discussion of the various measures of the underlying partisanship of a district, see Matthew S. Levendusky et al., Measuring District-Level Partisanship with Implications for the Analysis of U.S. Elections, 70 J. Pol. 736, 736-38 (2008).

(12.) SEE DAVE'S REDISTRICTING, (last visited June 6, 2016).

(13.) See BILL BISHOP, THE BIG SORT: WHY THE CLUSTERING OF LIKE-MINDED AMERICA IS TEARING US APART 5-15 (2008) (explaining the Big Sort phenomenon in politics); see also Jesse Sussell, New Support for the Big Sort Hypothesis: An Assessment of Partisan Geographic Sorting in California, 1992-2010, 46 POL. SCI. & POL. 768 (2013) (providing new empirical support for the Big Sort hypothesis and addressing the authors' critique in Samuel J. Abrams & Morris P. Fiorina, "The Big Sort" That Wasn't: A Skeptical Reexamination, 45 POL. SCI. & POL. 203 (2012)); Wendy K. Tam Cho et al., Voter Migration and the Geographic Sorting of the American Electorate, 103 ANNALS ASS'N AM. GEOGRAPHERS 1 (2013) (analyzing voter migration to areas populated by copartisans).

(14.) The converse belief--i.e., the belief that gerrymandering of districts leads to increased polarization--is common. See Norman J. Ornstein, The Pernicious Effects of Gerrymandering, Am. ENTERPRISE Inst. (Dec. 4, 2014, 9:02 AM), /publication/pernicious-effects-gerrymandering. Voter and legislator polarization, however, is not reduced in cases where district boundaries do not matter, such as the Senate, at-large House districts, or in randomly drawn districts. Thus, gerrymandering might not be a direct cause of polarization. See Michael J. Barber & Nolan McCarty, Causes and Consequences of Polarization, in SOLUTIONS TO POLITICAL POLARIZATION IN AMERICA 15,27-29 (Nathaniel Persily ed., 2015).

(15.) See Andrew Gelman & Gary King, Estimating the Electoral Consequences of Legislative Redistricting, 85 J. Am. St at. Assoc. 274,281 (1990).

(16.) See Greg Giroux, Republicans Win Congress as Democrats Get Most Votes, BLOOMBERG (Mar. 18, 2013 5:00 PM PDT), get-most-votes.

(17.) U.S. CENSUS BUREAU, STATISTICAL ABSTRACT OF THE UNITED STATES 261 tbl.418 (2012); Nicholas O. Stephanopoulos & Eric M. McGhee, Partisan Gerrymandering and the Efficiency Gap, 82 U. Chi. L. Rev. 831, 836 (2015) (calculating partisan-asymmetry trends over time from 1972 to 2012 and concluding that "[t]he severity of today's gerrymandering is ... unprecedented in modern times"); Carl Klarner, State Partisan Balance Data, 1937-2011, Harv. Data VERSE VI (2014), (documenting one-party rule and updating Carl Klarner, The Measurement of the Partisan Balance of State Government, 3 St. POL.&Pol'yQ. 309, 309-15 (2003)).

(18.) See Edward R. Tufte, The Relationship Between Seats and Votes in Two-Party Systems, 67 Am. Pol. Sci. Rev. 540, 542-43 (1973). For example, in a two-party system, it is theoretically possible for one political party to win forty-nine percent of the vote in every district, yet not win a single delegate. Although such an extreme case is highly improbable, strong deviations from proportionality are nevertheless an inherent risk of a winner-take-all district system. From a democratic standpoint, a central question is how to avoid the most extreme distortions. Actual nongerrymandered outcomes are considerably less distorted than the extreme hypothetical scenario described above.

(19.) 136 S.Ct. 450 (2015).

(20.) 136 S.Ct. 1301 (2016).

(21.) ANDREW HACKER, CONGRESSIONAL DISTRICTING: THE ISSUE OF EQUAL REPRESENTATION 40 (rev. ed. 1964) (describing the origin of single-member districts in the United States); Tory Mast, The History of Single Member Districts for Congress: Seeking Fair Representation Before Full Representation, FairVote (1995), Congress first required single-member districts in 1842. See Act of June 25, 1842, ch. 47, [section] 2, 5 Stat. 491, 491 ("[N]o one district elect[s] more than one Representative."). Congress has continued to pass apportionment acts requiring single-member districts. See Act of July 14, 1862, ch. 170, 12 Stat. 572; Act of Feb. 2, 1872, ch. 11, [section] 2,17 Stat. 28,28; Act of Feb.25,1882, ch. 20, [section] 3,22 Stat. 5,6; Act of jan. 16, 1901, ch. 93, [section] 3, 31 Stat. 733, 734; Act of Aug. 8, 1911, Pub. L. No. 62-5, [section] 3, 37 Stat. 13, 14; Act of Dec. 14, 1967, Pub. L. No. 90-196, 81 Stat. 581 (codified as amended at 2 U.S.C. [section] 2c (2014)).

(22.) Similar problems exist at the level of state legislatures. The analysis described in this Article is also applicable to evaluating state-level districting results.

(23.) U.S. Const, art. I, [section] 2.

(24.) U.S. Const, art. I, [section] 4.

(25.) Guidance Concerning Redistricting Under Section 5 of the Voting Rights Act, 76 Fed. Reg. 7470 (Feb. 9, 2011); Redistricting Information, U.S. Dep't Just. (Aug. 6, 2015), (containing information regarding sections 2 and 5 of the Voting Rights Act); State-by-State Redistricting Procedures, BALLOTPEDIA, (last visited June 6, 2016) (describing the legislative process for redistricting in each state); Where Are the Lines Drawn, All About REDISTRICTING, (last visited June 6, 2016) (describing political, geographical, and legal constraints on redistricting).

(26.) Tufte, supra note 18, at 542-43.

(27.) Sam Wang, Opinion, The Great Gerrymander of 2012, N.Y. Times (Feb. 2, 2013),

(28.) LEVITT, supra note 11, at 57-58.

(29.) Sam Wang, Opinion, Let Math Save Our Democracy, N.Y. Times (Dec. 5, 2015), YR7AdU.


(31.) See, e.g., THOMAS E. MANN & NORMAN J. ORNSTEIN, IT'S EVEN WORSE THAN IT LOOKS: HOW THE AMERICAN CONSTITUTIONAL SYSTEM COLLIDED WITH THE NEW POLITICS OF EXTREMISM 143-47 (2012) (addressing the partisan consequences of gerrymandering and discussing mechanisms for curbing extremism); Thomas E. Mann & Norman J. Ornstein, Opinion, Let's Just Say It: The Republicans Are the Problem, Wash. Post (Apr. 27, 2012),

(32.) See Litigation in the 2010 Cycle, All About Redistricting, (last visited June 6, 2016) (listing redistricting challenges pending before the Supreme Court).

(33.) 478 U.S. 109(1986).

(34.) 541 U.S. 267 (2004).

(35.) 548 U.S. 399 (2006).

(36.) 478 U.S. at 125.

(37.) Id. at 127 (plurality opinion) (agreeing with the district court that plaintiffs must prove both discriminatory intent and effect).

(38.) Id. at 122-23 (majority opinion); Vieth, 541 U.S. at 314 (Kennedy, J., concurring in the judgment) ("[T]hese allegations involve the First Amendment interest of not burdening or penalizing citizens because of their participation in the electoral process, ... their association with a political party, or their expression of political views.").

(39.) 429 U.S. 252,266-68 (1977).

(40.) 541 U.S. at 271 (plurality opinion).

(41.) Id. at 270.

(42.) Id.

(43.) Id. at 286.

(44.) Id. at 339 (Stevens, J., dissenting).

(45.) Id. at 314 (Kennedy, J., concurring in the judgment).

(46.) Id.

(47.) Davis v. Bandemer, 478 U.S. 109,133 (1986).

(48.) Mat 143.

(49.) 541 U.S. at 279, 281 (plurality opinion).

(50.) See id. at 271; id. at 306 (Kennedy, J., concurring in the judgment); id. at 317 (Stevens, J., dissenting); id. at 343 (Souter, J., dissenting); id. at 355 (Breyer, J., dissenting).

(51.) Id. at 306 (Kennedy, J., concurring in the judgment) ("I would not foreclose all possibility of judicial relief if some limited and precise rationale were found....").

(52.) See supra note 50 and accompanying text.

(53.) League of United Latin Am. Citizens (LULAC) v. Perry, 548 U.S. 399,409 (2006).

(54.) See id. at 420 (presuming that a partisan gerrymandering challenge could be litigated based on actual election results but not "in a hypothetical state of affairs," and stating that "[w]ithout altogether discounting its utility in redistricting planning and litigation, I would conclude asymmetry alone is not a reliable measure of unconstitutional partisanship"); id. at 473 n.11 (Stevens, J., concurring in part, dissenting in part) (describing asymmetry as one of eight criteria for determining effects-based violations); id. at 468 n.9 (describing the symmetry standard as "a helpful (though certainly not talismanic) tool"); id. at 466 (finding that the challenged plan was "inconsistent with the symmetry standard" and asserting that the "symmetry standard... is undoubtedly 'a reliable standard' for measuring a 'burden ... on the complainants' representative rights'" (quoting id. at 418 (majority opinion)); id. at 483 (Souter, J., concurring in part, dissenting in part) (explaining that he "do[es] not rule out the utility of a criterion of symmetry" as the Court's "interest in exploring this notion is evident"). For a further review of these statements in LULAC, see Bernard Grofman & Gary King, The Future of Partisan Symmetry as a Judicial Test for Partisan Gerrymandering After LULAC v. Perry, 6 ELECTION L.J. 2,4-5 (2007).

(55.) 541 U.S. 267,306-07 (Kennedy, J., concurring in the judgment).

(56.) Davis v. Bandemer, 478 U.S. 109,155 (O'Connor,)., concurring

in the judgment).

(57.) 541 U.S. at 282 (plurality opinion).

(58.) Brief for Appellants at 20, Vieth, 541 U.S. 267 (No. 02-1580).

(59.) Vieth, 541 U.S. at 360 (Breyer, J., dissenting).

(60.) Id. at 289 (plurality opinion).

(61.) The zone-of-chance concept is a way to express the concept of significance testing in statistics. Statisticians calculate how far a measurement, such as the number of seats won by a party in a given election, is likely to stray from the expected average. The yardstick for the amount strayed is the "standard deviation," a quantity denoted by the Greek letter sigma ([sigma]). In this Article, I define the zone of chance as a region within which chance outcomes would fall ninety-five percent of the time and outside the region five percent of the time. Statistics texts refer to this as a "p<0.05" or "[alpha]<0.05" standard. The size of the zone of chance is some multiple of sigma, which can be calculated, and is always at least 1.6 times sigma for a bell-shaped curve and 1.75 times sigma for a f-distribution. See also Wang, supra note 29. See generally RICHARD LOWRY, Tests of Statistical Significance: Three Overarching Concepts, in CONCEPTS AND Applications of Inferential Statistics (2000), /ch7ptl.html (providing an introduction to the concept of significance testing using confidence intervals).

(62.) Vieth, 541 U.S. at 296 (plurality opinion).

(63.) Id. at 353 (Souter, J., dissenting).

(64.) As an example, racial gerrymandering is judged one district at a time, not on a statewide basis. See, e.g., Ala. Legislative Black Caucus v. Alabama, 135 S. Ct. 1257, 1265 (2015); Miller v. Johnson, 515 U.S. 900, 916 (1995); Shaw v. Reno, 509 U.S. 630, 649 (1993). Even more broadly, the word "gerrymander" is colloquially used to describe a range of partisan offenses, including polarization of voters. Such overbroad usage dates back at least a hundred years. See Griffith, supra note 1, at 26-27. In this Article, the term is restricted to the narrower sense of using district boundaries to obtain an advantage for a candidate, faction, or party.

(65.) Mathematically, this can be stated as follows. If party A gets fraction V of the total two-party vote, and all districts on both sides will be split 60-40, then F, the fraction of A-favoring districts, must satisfy 0.6F+0.4(1-F)=V. Furthermore, if V>0.5, i.e., party A wins the popular vote, then F>0.5, i.e., the number of A-favoring districts must also be a majority. This principle is generally true, and is limited only by the fact that for a finite number of districts, the margins of the individual districts would not be precisely 60-40.

(66.) Vieth, 541 U.S. at 298 (plurality opinion); see Gaffney v. Cummings, 412 U.S. 735, 752-54 (1973).

(67.) See Voting Rights Act of 1965, Pub. L. No. 89-110, [section] 2, 79 Stat. 437, 437 (codified as amended at 52 U.S.C. [section] 10301 (2014)). For the establishment of majority-minority districts, see Thornburg v. Gingles, 478 U.S. 30, 46-51 (1986); and Bartlett v. Strickland, 556 U.S. 1,7-22 (2009).

(68.) See Stephen Ansolabehere & Maxwell Palmer, A Two Hundred-Year Statistical History of the Gerrymander 13-15 (May 16, 2015) (unpublished manuscript),

(69.) Election 2012: Michigan, N.Y. Times, /michigan (last visited June 6,2016); Michigan Election Results, N.Y. TIMES (Dec. 17, 2014, 12:28 PM),

(70.) 541 U.S. 267,290-91 (2004) (plurality opinion) (quoting Davis v. Bandemer, 478 U.S. 109, 161 (1986) (Powell, J., concurring in part and dissenting in part)).

(71.) See id. at 291.

(72.) Id. at 284.

(73.) See Bandemer, 478 U.S. at 143 (plurality opinion).

(74.) See, e.g., Paul Gronke & J. Matthew Wilson, Competing Redistricting Plans as Evidence of Political Motives: The North Carolina Case, 27 Am. Pol. Q. 147 (1999); Seth C. McKee et al., The Partisan Impact of Congressional Redistricting: The Case of Texas, 2001-2003, 87 Soc. SCI.Q. 308 (2006).

(75.) Jowei Chen & Jonathan Rodden, Unintentional Gerrymandering: Political Geography and Electoral Bias in Legislatures, 8 Q.J. Pol. SCI. 239, 248 (2013) [hereinafter Chen & Rodden, Unintentional Gerrymandering]; see also Jowei Chen & Jonathan Rodden, Cutting Through the Thicket: Redistricting Simulations and the Detection of Partisan Gerrymanders, 14 ELECTION L.J. 331, 332 (2015) [hereinafter Chen & Rodden, Cutting Through the Thicket); Jowei Chen & Jonathan Rodden, Report on Computer Simulations of Florida Congressional Districting Plans (Feb. 15, 2013) (unpublished manuscript) (on file with author); Jowei Chen & Jonathan Rodden, Supplemental Report on Partisan Bias in Florida's Congressional Redistricting Plan (Oct. 21, 2013) (unpublished manuscript) (on file with author).

(76.) Use of real results is also consistent with League of United Latin American Citizens (LULAC) v. Perry, 548 U.S. 399, 420 (2006) (expressing wariness of identifying asymmetry-based results that would occur in a hypothetical circumstance).

(77.) See infra Part II.A.

(78.) 541 U.S. 267, 287-88 (2004) (plurality opinion) (describing and rejecting appellants' proposed effects standard).

(79.) Id.

(80.) See generally Grofman & King, supra note 54, at 4 (explaining partisan symmetry concept).

(81.) See supra note 54 and accompanying text.

(82.) 541 U.S. at 279 (plurality opinion).

(83.) Id. at 281.

(84.) See supra note 51 and accompanying text.

(85.) See Grofman & King, supra note 54, at 4.

(86.) A failure rate of 2 out of 34, or 6%, may be considered acceptable, when one considers the following comparison: in the history of the United States, the popular-vote winner has failed to win the presidency in 4 out of 57 elections, a 7% rate. See United States Presidential Election Results, Dave Leip's Atlas U.S. PRESIDENTIAL ELECTIONS (2012), (to locate, search presidential election results by year and compare popular vote to electoral vote). However, presidential elections rely on fixed state boundaries. Maintaining representative performance in legislative elections is vulnerable to variations in where and how district boundaries are drawn.

(87.) See Haas, supra note 29, at 3, 31,46,53,66.

(88.) Jess Bravin, Supreme Court Revives Challenge to North Carolina Redistricting, WALL St. J. (Apr. 20, 2015),; Griff Palmer & Michael Cooper, How Maps Helped Republicans Keep an Edge in the House, N.Y. TIMES (Dec. 14, 2012),

(89.) See Delia Baldassarri & Andrew Gelman, Partisans Without Constraint: Political Polarization and Trends in American Public Opinion, 114 Am. J. SOC. 408, 423-26 (2008) (showing that across a range of economic, civil rights, and moral issues, correlations between issue partisanship and party identification are positive and increasing over time).

(90.) See Ariz. Sec'y of State, State of Arizona Official Canvass: 2012 General Election 4-6 (2012),

(91.) Vieth v. Jublirer, 541 U.S. 267,298 (2004) (plurality opinion).

(92.) Id. at 312-13 (Kennedy, J., concurring in the judgment).

(93.) On the majority-minority district principle, see Voting Rights Act of 1965, Pub. L. No. 89-110, [section] 2, 79 Stat. 437, 437 (codified as amended at 52 U.S.C. [section] 10301 (2014)); Thornburg v. Gingles, 478 U.S. 30, 46-51 (1986); and Bartlett v. Strickland, 556 U.S. 1,7-22 (2009).

(94.) Redistricting and You: How New York State's Approved Redistricting Lines Compare with Old Districts, Ctr. FOR URBAN Res., /map.html (last visited June 6,2016).

(95.) See Nathaniel Persily, The Promise and Pitfalls of the New Voting Rights Act, 117 Yale L.J. 174, 235-45 (2007).

(96.) See 52 U.S.C. [section] 10301(b) ("[N]othing in this section establishes a right to have members of a protected class elected in numbers equal to their proportion in the population.").

(97.) See Gingles, 478 U.S. at 43,75; see also Johnson v. De Grandy, 512 U.S. 997, 1000 (1994).

(98.) 512 U.S. at 1000.

(99.) See Gingles, 478 U.S. at 74-77 (describing near-proportional legislative representation of black voters as evidence of their ability to elect their preferred representatives).

(100.) In this plot, the black line indicates proportionality and is a straight line drawn from zero vote share and zero seat fraction to 100% vote share and 100% seat fraction. The seats/votes curve is calculated by resampling to build "fantasy delegations," see infra Part II.A.3, and is approximated by the mathematical function that is the area under a bell-shaped curve whose average is 50% vote share, and whose standard deviation is 14% vote share.

(101.) Proportional representation is achieved only in systems where it is enforced specifically and directly. For example, in Israel, members of the national legislative body, the Knesset, are assigned so that the number of a party's seats is proportional to the fraction of its popular vote. Basic Law: The Knesset, 5718, [section] 4, 180 LSI 18 (19872003). Such a system embodies a legislature-centered form of the "one person, one vote" principle: Each citizen's party preference is reflected proportionally at the national level.

(102.) Reynolds v. Sims, 377 U. S. 533,565-66 (1964).

(103.) See Wang, supra note 29.

(104.) The zone-of-chance concept is a way to express the concept of significance testing in statistics. Statisticians calculate how far a measurement, such as the number of seats won by a party in a given election, is likely to stray from the expected average. In this Article, I define the zone of chance as a region within which chance outcomes would fall 95% of the time and outside the region 5% of the time. Statistics texts refer to this as a "p<0.05" or "[alpha]<0.05" standard. See Lowry, supra note 61; see also Wang, supra note 29.

(105.) Vieth v. Jubelirer, 541 U.S. 267, 312-13 (2004) (Kennedy, J., concurring in the judgment).

(106.) It must be noted that the simplified formula for sigma described in this paragraph is a substantial overestimate of real-life situations because districting generates a mixture of more and less closely-contested districts, and only close contests contribute to uncertainty. To estimate the true value of sigma, which is typically smaller, a more sophisticated approach is required, as detailed in Part I1.A.3 below.

(107.) For example, if all N races are perfect toss-ups, then they behave like coin tosses, and according to the laws of probability, the standard deviation of the seat outcome--a measure of variation often referred to as sigma, or [sigma]--is 0.5*[square root of N]. Thus if political parties A and B compete in a state that is composed of sixteen congressional districts, all of which are closely contested, then each party can expect to get eight seats on average. Sigma for the specific case of all-close-races is 0.5*[square root of 16] = 2 seats, suggesting that each party would typically get six to ten seats. For an approximate formula that applies to a wider range of situation, see infra note 117.

(108.) This can be done by using all 435 House race outcomes. For a state X with N districts, you would calculate the total popular vote across all Ndistricts, then pick Nraces from around the country at random and add up their vote totals. If their vote total matches X's actual popular vote within 0.5%, score it as a comparable simulation. See, e.g., Wang, supra note 27.

(109.) It is possible to explore the properties of this simulation procedure by giving it a variety of hypothetical nationwide distributions of districts as starting data. These hypothetical scenarios reveal that the "fantasy delegation" procedure has important features that are required of a detector of partisan asymmetry. First, for a symmetric distribution of congressional districts, i.e., a scenario in which Democrat-dominated districts are no more packed than Republican-dominated districts, fantasy delegations are typically majoritarian, awarding more representatives to the party that receives more votes. Second, the fantasy delegations have the same natural variation in partisan composition as the actual nationwide distribution of state delegations, as measured by standard deviation. Third, when the nationwide distribution of districts has asymmetry, for instance containing a number of districts that are very packed with one party (as is the case in real life for Democrats), the fantasy delegations show a bias toward the other party, a phenomenon that is well analyzed. See Chen & Rodden, Unintentional Gerrymandering, supra note 75.

(110.) The MATLAB software is available at Sam Wang, Gerrymandering, GitHub, (last visited June 6,2016).

(111.) Tim Dickinson, How Republicans Rig the Game, ROLLING STONE (Nov. 11, 2013),; Giroux, supra note 16; Olga Pierce et al, How Dark Money Helped Republicans Hold the House and Hurt Voters, ProPublica (Dec. 21, 2012, 2:36 PM),

(112.) Statewide vote totals may include some races that are uncontested. In these districts, it is not known how the voters would have decided if they had an alternative choice. In order to address this, it may also be necessary to assign those voters assuming a split other than 100%-0%. One established approach is to assume a 75%-2S% split. See Andrew Gelman & Gary King, A Unified Method of Evaluating Electoral Systems and Redistricting Plans, 38 Am. J. POL. SCI. 514, 550 (1994). Generally, this assumption does not affect the outcome of the tests in this Article.

(113.) See Dickinson, supra note 111; Giroux, supra note 16; Pierce et al., supra note 111.

(114.) In Arizona, small shifts in voting in either the second or ninth district would have altered the overall outcome to near-neutrality. See Ariz. Sec'y of State, supra note 90. Texas is a complex case in which redistricting was constrained by multiple factors favoring both parties, including the establishment of multiple ability-to-elect districts. See Redistricting in Texas, BALLOTPEDIA, (last visited June 6, 2016).

(115.) Aaron Blake & Chris Cillizza, The Top 10 States to Watch in Redistricting, WASH. POST: POL. BLOG (Mar. 18, 2011), redistricting/2011/03/18/ABju9Ar_blog.html.

(116.) A difference of Delta = 1 or more in a dys-proportional direction occurs in approximately 16% of cases. A difference of Delta = 2 or more occurs in approximately 2.3% of cases. A difference of Delta = 3 or more occurs in approximately 0.13% of cases. These values are for Analysis 1, which uses a bell-shaped curve, the usual assumption for statistical testing. Analyses 2 and 3 use the t-distribution, which gives slightly different values. I define the quantity Delta ([DELTA]) as the difference from average expectations, divided by sigma.

(117.) These values are approximated reasonably well by the formula sigma = 0.52 x [square root of (s x (N - s) / N)], where N is the number of a state's congressional districts and s is the average number of seats won in that state by either major party in computer simulations. The principal difference from the "all toss-ups" example is the appearance of a factor of 0.52, which arises from the fact that some districts are competitive, and some are not; this factor fell within a narrow range of 0.50-0.53 between 2008 and 2014.

(118.) See 2012 REDMAP Summary Report, REDMAP: REDISTRICTING MAJORITY PROJECT (Jan. 4, 2013,9:23 AM),

(119.) See Redistricting in Texas: Redistricting After the 2010 Census, BALLOTPEDIA, https://ballotpedia.Org/Redistricting_in_Texas#Redistricting_after_the_2010_census (last visited June 6,2016).

(120.) See Redistricting in Arizona: Redistricting After the 2010 Census, BALLOTPEDIA, https://ballotpedia.Org/Redistricting_in_Arizona#Redistricting_after_the_2010_census (last visited June 6,2016).

(121.) See Redistricting in California: Redistricting After the 2010 Census, BALLOTPEDIA, https://balIotpedia.Org/Redistricting_in_California#Redistricting_after_the_2010_census (last visited June 6, 2016).

(122.) HAAS, supra note 30, at 72.

(123.) A theoretical symmetric distribution of districts would, on average, give a delegation that is 79% Democratic. For a symmetrically distributed distribution of districts whose two-party vote share has standard deviation SD, the expected fraction of seats S for a given vote share V is normcdf((V-0.5)/SD), where normcdf is the integral of a bell-shaped normal curve with mean 0 and width parameter 1. For non-dys-proportional states in 2012, SD = 0.15, comparable to longstanding findings for seats/votes curves. See GRAHAM GUDGIN & PETER J. TAYLOR, SEATS, VOTES, AND THE SPATIAL ORGANISATION OF ELECTIONS 20-31(1979).

(124.) See Perry v. Perez, 132 S. Ct. 934, 939-40 (2012) (per curiam). For a fuller accounting of the lengthy redistricting battle in Texas, see Redistricting in Texas, supra note 114.

(125.) See HAAS, supra note 30, at 57-59.


(127.) Id.


(129.) Texas Election 2012, N.Y. Times, (last visited June 6,2016).

(130.) Chen & Rodden, Cutting Through the Thicket, supra note 75, at 335-38.

(131.) Id. at 338.

(132.) FLA. CONST, art. III, [section][section] 16(c), 20-21.

(133.) League of Women Voters of Fla. v. Detzner, 172 So. 3d 363,371-72 (Fla. 2015).

(134.) Monica Davey, One-Party Control Opens States to Partisan Rush, N.Y. TIMES (Nov. 22, 2012),; State Government Control Since 1938, N.Y. TIMES (Nov. 22, 2012),

(135.) Davey, supra note 134; State Government Control Since 1938, supra note 134.

(136.) Davey, supra note 134; State Government Control Since 1938, supra note 134.

(137.) HAAS, supra note 30, at 74.

(138.) Chen & Rodden, Unintentional Gerrymandering, supra note 75, at 260-64 (calculating the biases associated with simulated redistricting using compactness principles across the fifty states).

(139.) See GUDGIN & TAYLOR, supra note 123, at 20-31.

(140.) This effect is consistent with previous work. See Stephanopoulos & McGhee, supra note 17, at 873 fig.5.


(142.) Compare id. at 2, 9, 13, 19, 21, 36, 37, 40, 44, 48, 50 (giving House election outcomes in 2012 for the states listed in Table 1 above), with HAAS, supra note 30, at 2,12, 18, 27, 30, 46, 47, 52, 63, 65 (giving House election outcomes in 2014 for the states listed in Table 1 above).

(143.) Wang, supra note 29.

(144.) LEVITT, supra note 11, at 58.

(145.) Because members of both major parties get packed into districts in a partisan gerrymander, individual members of the opposing party may acquiesce or even be complicit in the process. See, e.g., League of United Latin Am. Citizens (LULAC) v. Perry, 548 U.S. 399, 418 (2006) ("[A] number of line-drawing requests by Democratic state legislators were honored."). In other words, a single-district gerrymander can favor one party even as a partisan gerrymander favors the other party. For this reason, the use of intent as a standard for gerrymandering should distinguish between district-level and party-level motivations.

(146.) Chen & Rodden, Unintentional Gerrymandering, supra note 75, at 241.

(147.) BISHOP, supra note 13, at 5-15.

(148.) Mitchell N. Berman, Managing Gerrymandering, 83 TEX.L. REV. 781,781 (2005).

(149.) RICHARD LOWRY, t-Test for the Significance of the Difference Between the Means of Two Independent Samples, in CONCEPTS AND APPLICATIONS OF INFERENTIAL STATISTICS, supra note 62,

(150.) The mean-median difference was originally suggested as a measure of partisan gerrymandering in Michael D. McDonald & Robin E. Best, Unfair Partisan Gerrymanders in Politics and Law: A Diagnostic Applied to Six Cases, 14 ELECTION L.J. 312, 312(2015).

(151.) The presence of uncontested races reduces the value of the mean-minus-median statistic. In those cases, the partisan breakdown is not known with accuracy. Consider the example of a twenty-district state. Residents of an uncontested district would have voted at a rate of 80% for their party, instead of the nominal 100%. If their district were drawn differently, the appropriate mean for comparison would be based on the 80% figure and shift the overall mean by 1%.

(152.) The mean-median difference is a simple and old measure of "skewness," a statistical term for asymmetry. See G. UDNY YULE & M.G. KENDALL, AN INTRODUCTION TO THE THEORY OF STATISTICS 162-63 (14th ed. 1968); David P. Doane & Lori E. Seward, Measuring Skewness: A Forgotten Statistic?, J. STAT. EDUC., July 2011, at 9-10; Karl Pearson, Contributions to the Mathematical Theory of Evolution--II: Skew Variation in Homogeneous Material, PHIL. TRANSACTIONS Royal Soc'y, 1895, at 343, 374-76.

(153.) Haas, supra note 30, at 53. Democratic two-party share is defined as the number of Democratic votes divided by the sum of Democratic and Republican votes, expressed as a percentage.

(154.) The level of statistical significance is calculated using Test 3 in Part III.A below, and Student's 1-distribution. For the original calculation of the f-distribution, see Student, The Probable Error of a Mean, 6 BIOMETRIKA 1, 1 (1908); see also LOWRY, supra note 149 (explaining Student's 1-distribution).

(155.) In Ohio, one race (the eleventh district) was uncontested and won by a Democrat, Marcia Fudge. Haas, supra note 30, at 48.

(156.) This [[sigma].sub.1] can be calculated according to the formula for sigma described above, see supra note 107 and accompanying text, or by numerical simulation, see supra Part II.A.3.

(157.) See supra Part II.B.

(158.) See LOWRY, supra note 149.

(159.) This is the mean-median test described by the Author in Wang, supra note 29, and in McDonald & Best, supra note 150, at 321-29.

(160.) See Paul Cabilio & Joe Masaro, A Simple Test of Symmetry About an Unknown Median, 24 CAN. J. STAT. 349, 352 (1996) (locating the standard deviation as the square root of the quantity ao2(F) in Table 1); Tian Zheng & Joseph L. Gastwirth, On Bootstrap Tests of Symmetry About an Unknown Median, 8 J. DataSci. 397, 400-01 (2010).

(161.) See GEORGE W. SNEDECOR & WILLIAM G. COCHRAN, STATISTICAL METHODS 251-52 (8th ed. 1989); Karl Pearson, On the Criterion That a Given System of Deviations from the Probable in the Case of a Correlated System of Variables Is Such That It Can Be Reasonably Supposed to Have Arisen from Random Sampling, 5 PHIL. Mag. SERIES 157, 163-67 (1900) (describing the original mathematical derivation of the chi-square statistic, the practical use of which, for purposes of analyzing redistricting, is better accomplished using the online calculator at Sam Wang, Gerrymandering Demo, PRINCETON Univ., (last visited June 6, 2016)).

(162.) A typical level of statistical significance is to set the threshold for Delta so that chance would give the observed result 5% of the time or less. Whether this occurs depends on Delta, which increases in proportion to the square root of the number of districts. See supra note 116 and accompanying text. Delta is evaluated by comparison with significance values for the f-distribution. For Tests 2 and 3, statistical significance is typically reached when Delta exceeds 1.75.

(163.) The three tests proposed here address the overall apportionment plan, but do not cover the case of individual self-dealing in single districts. Local laws may provide additional constraints. For example, the current congressional districts in Florida do not violate my three tests. Nonetheless, the Florida Supreme Court has held that the map violates the Florida Constitution redistricting provisions. See League of Women Voters v. Detzner, 172 So. 3d 363, 427 (Fla. 2015) (relying on Fla. Const, art. Ill, [section] 20(a) which mandates that "[n]o apportionment plan or individual district shall be drawn with the intent to favor or disfavor a political party or an incumbent"). This stricter standard extends a mandate for competitive races to the level of single districts.

(164.) Harris v. Ariz. Indep. Redistricting Comm'n, 993 F. Supp. 2d 1042 (D. Ariz. 2014) (per curiam), aff'd, 136 S. Ct. 1301 (2016).

(165.) See Shapiro v. McManus, 136 S. Ct. 450,456 (2015).

(166.) See Lampi Collection, supra note 4 (listing election results). For the calculation of Test 1, each district election is used as one data value. For the number of districts, see Griffith, supra note 1, at 62.

(167.) Lampi Collection, supra note 4.

(168.) Id. In that election, multimember districts of unequal population were allowed. Equipopulation districts were not required until the Supreme Court held that malapportionment claims were justiciable in Baker v. Carr, 369 U.S. 186 (1962), and later developed the one person, one vote standard in Reynolds v. Sims, 377 U. S. 533 (1964). For a history of the one person, one vote standard, see Samuel Issacharoff et al., The Law of Democracy: Legal Structure of the Political Process 126-213 (4th ed. 2012).

(169.) ARIZ. CONST, art. 4, pt. II, [section] 1.

(170.) Brief for Appellants at i, Harris v. Ariz. Indep. Redistricting Comm'n, 136 S. Ct. 1301 (2016) (No. 14-232), 2015 WL 5261558.

(171.) Harris v. Arizona Indep. Redistricting Comm'n, 993 F. Supp. 2d 1042, 1047 (D. Ariz. 2014) (per curiam), aff'd, 136 S. Ct. 1301 (2016).

(172.) Id. at 1046.

(173.) Harris, 136 S. Ct. at 1305.

(174.) Id. at 1310.

(175.) Brief for Appellants, supra note 170, at 17.

(176.) Ariz. Sec'y of State, State of Arizona Official Canvass: 2014 General Election 2-6 (2014), vass2014GE.pdf. Two-party vote share is defined in the same way as for House districts: the number of votes for one party divided by the sum of the two parties' votes, expressed as a percentage. See supra note 153 and accompanying text.

(177.) Brief for Petitioners at 12, 35-39, Shapiro v. McManus, 136 S. Ct. 450 (2015) (No. 14990), 2015 WL 4720269.

(178.) Benisek v. Mack, 584 F. App'x 140, 141 (4th Cir. 2014), rev'd sub nom. Shapiro v. McManus, 136 S. Ct. 450 (2015).

(179.) Shapiro, 136 S. Ct. at 456.

(180.) KAREN L. HAAS, CLERK OF THE HOUSE OF REPRESENTATIVES, STATISTICS OF THE CONGRESSIONAL ELECTION OF NOV. 2, 2010, at 22 (June 3, 2011), /member_info/electionlnfo/2010election.pdf.

(181.) Haas, supra note 30, at 27-28.

(182.) Maryland Election Results 2014, N.Y. Times (Dec. 17, 2014, 12:28 PM),

(183.) The standard deviation is the square root of the variance.

(184.) For a lower one-tailed test at significance level p<0.05, the lower bound of the zone of chance is equal to (national standard deviation)*. [square root of (2.167/(Af-1))]. See Chi-Square Distribution Table (n.d.), (giving a table of critical values above which the chi-square score is statistically significant); Chi-Square Test for the Variance, ENGINEERING STATISTICS HANDBOOK, (last visited June 6, 2016) (describing how the chi-square statistic is calculated). It should also be noted that the chi-square test assumes normally distributed vote shares. An additional test, the Ansari-Bradley test, does not make this assumption, and still identifies 2012 and 2014 (but not 2004) as being statistically significant departures from national Democratic districts. See A.R. Ansari & R.A. Bradley, Rank-Sum Tests for Dispersions, 31 Annals MATHEMATICAL St AT. 1174, 1175-76 (1960) (describing a test statistic termed "W" which uses rankings on a sorted list as a means of testing for differences between groups).

(185.) Without partisan intent, the Maryland standard deviation would still be expected to fall outside the zone of chance in five percent of cases--one in twenty. Maryland's 2004 congressional delegation was within the zone of chance by Test 1, indicating that the result of Test 3 is a chance result, i.e., a "false positive."

(186.) A version of this software is available on Github at Wang, supra note 110, and a web browser-based version is available at Wang, supra note 161.

(187.) Vieth v. Jubelirer, 541 U.S. 267, 315-16 (2004) (Kennedy, J., concurring in the judgment).

(188.) See Davis v. Bandemer, 478 U.S. 109, 129 (1986); Vieth, 541 U.S. at 284-86 (plurality opinion) (reviewing the difficulty of meeting a standard of "predominant intent").

(189.) Micah Altman et al., Revealing Preferences: Why Gerrymanders Are Hard to Prove, and What to Do About It 11-36 (Mar. 22, 2015) (unpublished manuscript), (enumerating the strengths and weaknesses of different approaches to evaluating partisan gerrymanders).

(190.) League of United Latin Am. Citizens (LULAC) v. Perry, 548 U.S. 399, 417-18 (2006) (citing the honoring of Democratic state legislators' requests as indicating that partisan gain was not the redistricters' sole motivation).

(191.) Id. at 418.

(192.) For a discussion of mixed partisan motivations, see note 145 above and accompanying text.

(193.) 553 U.S. 181,203-04 (2008).

(194.) In one recent example of a racial discrimination case, the Supreme Court held that demonstrating disparate impact was sufficient to prove discrimination, and that it was only necessary to demonstrate that the effect arose as a consequence of actions, as opposed to explicit racial intent. See Tex. Dep't of Housing & Comm. Aff. v. Inclusive Comms. Project, Inc., 135 S. Ct. 2507, 2517-19 (2015). The Court held that in light of the results-oriented statutory language in the Fair Housing Act, a showing of disparate impact was sufficient to warrant a remedy, even without discriminatory intent. Id. I argue that if gerrymandering has a sufficiently large effect on a party's supporters, such an injury should still be remedied even when redistricters are not motivated purely by partisan intent.

(195.) Justice Kennedy, joined by Justices Souter and Ginsburg, explained that with respect to a partisan gerrymandering claim, "such a challenge could be litigated if and when the feared inequity arose." LULAC, 548 U.S. at 420. Redistricting software is capable of using quantities such as the presidential vote share to estimate the partisan tendency of a hypothetical district. Redistricters use such measures to judge the likely outcome of a district and could use them as inputs to my three tests to evaluate a districting plan before it is implemented.

(196.) See Laws and Regulations, Cal. Citizens Redistricting Commission, (last visited June 6, 2016).

(197.) Cal. Const, art. XXI, [section] 2(c)(2).

(198.) Id. [section] 2(d).

(199.) See RAPHAEL J. SONENSHEIN, WHEN THE PEOPLE DRAW THE LINES: AN EXAMINATION OF THE CALIFORNIA CITIZENS REDISTRICTING COMMISSION 71-72 (2013), 13.pdf (reviewing academic and press analyses of the 2012 election, including increased competitiveness); Angelo N. Ancheta, Redistricting Reform and the California Citizens Redistricting Commission, 8 HARV. L. & POL'Y REV. 109, 135-36 (2014) (reviewing 2012 election outcomes).

(200.) Florida, ALL ABOUT REDISTRICTING, (last visited June 6,2016).

(201.) FLA. CONST, art. Ill, [section] 16.

(202.) W. [section][section] 20-21.

(203.) M. [section][section] 20(a), 21(a).

(204.) Id. [section][section] 16(c), 16(d).

Samuel S.-H. Wang, Faculty Associate, Program in Law and Public Affairs, and Professor, Princeton Neuroscience Institute, Princeton University. The Author thanks Dale Bratton, Michael Kimberly, Jonah Gelbach, Leslie Gerwin, Daniel Hemel, Michael P. McDonald, Rebecca Moss, Norm Ornstein, Nathaniel Persily, David Rosen, John Rodden, Kim Lane Scheppele, Michael Tiemann, and Maxim Zaslavsky for discussion and correspondence; Heather Gerken, Gary King, Matt McFarlane, and Nicholas Stephanopoulos for reading and commenting on the manuscript; David Hollander for assistance on references; Erik Beck, Carl Klarner, Matthew Harrison, and Maxim Zaslavsky for help with data; and the editors of the Stanford Law Review for helpful suggestions in the preparation of this Article. These tests of gerrymandering are available online for immediate use at
Table 1
Discrepancies Between Simulated and Actual Delegations
for the 2010-2014 House Elections


                Total   Democratic     Seats
                Seats   Vote Share     Seats

Arizona           8       46.74%         3
Florida          25#     42.53%#         6#
Illinois         19       54.96%         8
Indiana          9#      42.04%#         3#
Maryland          8       63.42%         6
Michigan         15#     47.87%#         6#
North            13       42.47%         4
Ohio             18#     44.75%#         5
Pennsylvania     19       48.41%         7
Texas            32#      37.51%         7
Virginia         11       44.94%         3
Wisconsin        8#      45.40%#         3#


                Simulated     SD      Difference
                 Average    (Sigma)     in SD

Arizona           3.18       0.78      R by 0.2
Florida           7.38#      1.31#    R by 1.1#
Illinois          11.15      1.23     R by 2.6# *
Indiana           2.53#      0.78#     D by 0.6
Maryland          6.13       0.78      R by 0.2
Michigan          6.35#      1.07#     R by 0.3
North             3.80       0.94      D by 0.2
Ohio              6.26#      1.14#    R by 1.1#
Pennsylvania      8.27       1.21     R by 1.1#
Texas             5.69#      1.30#    D by 1.0#
Virginia          3.87       0.89     R by 1.0#
Wisconsin         2.91#      0.77      Dby 0.1


                Total   Democratic   Democratic
                Seats   Vote Share     Seats

Arizona           9       45.60%         5
Florida          27#     50.00%#        10#
Illinois         18       55.40%         12
Indiana           9      45.80%#         2#
Maryland          8       65.46%         7
Michigan         14#     52.70%#         5
North            13       50.90%         4
Ohio             16#     47.90%#         4#
Pennsylvania     18       50.70%         5
Texas            36       39.90%        12#
Virginia         11       49.00%         3
Wisconsin         8       50.76%         3


                Simulated     SD      Difference
                 Average    (Sigma)     in SD

Arizona           2.96       0.76     D by 2.7# *
Florida          11.73#      1.33#    R by 1.3#
Illinois          10.04      1.11     D by 1.8#
Indiana           3.02#      0.76#    R by 1.3#
Maryland          6.11       0.72     D by 1.2#
Michigan          6.97       0.98#    R by 2.0# *
North             5.94       0.93     R by 2.1# *
Ohio              6.48#      1.03#    R bv 2.4# *
Pennsylvania      8.14       1.10     R by 2.9# *
Texas             8.68#      1.47#     Dby 2.3# *
Virginia          4.56       0.86     R by 1.8#
Wisconsin         3.64#      0.73      R by 0.9


                Democratic   Democratic   Simulated
                Vote Share     Seats       Average

Arizona           43.17%         4          2.94
Florida          49.02%#        10#        12.32#
Illinois          51.42%         10         9.14
Indiana          38.90%#         2#         2.03#
Maryland          58.14%         7          5.13
Michigan         50.88%#         5          6.96
North             45.31%         3          4.89
Ohio             40.94%#         4#         4.40#
Pennsylvania      46.07%         5          7.06
Texas             3935%         11#         8.65#
Virginia          44.84%         3          4.02
Wisconsin        47.20%#         3          3.36


                  SD      Difference
                (Sigma)     in SD

Arizona          0.75     D by 1.4#
Florida          2.59#     R by 0.9
Illinois         1.12      D by 0.8
Indiana          0.72#    R by 0.04
Maryland         0.76     D by 2.4# *
Michigan         0.98#    R by 2.0# *
North            0.92     R by 2.1# *
Ohio             0.98#     R by 0.4
Pennsylvania     1.09     R by 1.9#
Texas            1.43     D by 1.6#
Virginia         0.84     R by 1.2#
Wisconsin        0.73      R by 0.5

For 2010, 2012, and 2014, one million simulations were done for
each state, resampling was done from nationwide House election
returns for that year. The "SD (sigma)" column indicates
the value of sigma calculated from the simulations. Boldface text
indicates values of Delta (difference between simulation and
actual results exceeding one times sigma favoring either
party. Boldface underline indicates differences exceeding two
times sigma. Note the persistence of effects in 2014.

Note: Values of Delta (difference between
simulation and actual results exceeding one times
sigma favoring either party are indicated with #.

Note: Differences exceeding two
times sigma are indicated with *.

Table 2
Results of Three Tests for Partisan Asymmetry for
the Congressional Elections of 2012

                        Test 1 (Simulation)

                  Total   Simulated   Divided by
                  seats    Average      Sigma)

Arizona             9       2.96       D by 2.7#
Florida            27      11.73       R by 1.3
Illinois           18       10.04      D by 1.8#
Indiana             9       3.02       R by 1.3
Maryland            8       6.11       D by 1.2
Michigan           14       6.97       R by 2.0#
North Carolina     13       5.94       R by 2.1#
Ohio               16       6.48       R by 2.4#
Pennsylvania       18       8.14       R by 2.9#
Texas              36       8.68       D by 2.3#
Virginia           11       4.56       R by 1.8#
Wisconsin           8       3.64       R by 0.9

                          Test 2 (Lopsided Margins)

                  Democratic   Republican           Divided by
                     Win%         Win%      Sigma     Sigma)

Arizona             63.1%        66.6%      9.5%     D by 0.4
Florida             73.0%        67.4%      7.4%     R by 0.8
Illinois            66.2%        62.1%      4.9%     R by 0.8
Indiana             65.1%        59.5%      3.1%     R by 1.8#
Maryland            70.4%        66.5%        -          -
Michigan            74.4%        58.9%      4.9%     R by 3.2#
North Carolina      70.2%        57.5%      6.9%     R by 1.9
Ohio                80.2%        62.2%      7.5%     R by 2.4#
Pennsylvania        76.3%        59.5%      5.5%     R by 3.1#
Texas               71.4%        72.1%      4.5%     D by 0.2
Virginia            70.9%        58.8%      5.6%     R by 2.1
Wisconsin           68.9%        59.6%      3.8%     R by 2.4#

                     Test 3 (Skewed Districts)

                   Directly From Election Returns

                    Minus                  [DELTA]
                    Median             (Average-Median
                  Democratic             Divided By
                   Vote (%)    Sigma       Sigma)

Arizona             -0.5%      3.8%       D by 0.1
Florida              4.8%      3.8%       R by 1.2
Illinois             2.1%      3.1%       R by 0.7
Indiana              1.4%      2.1%       R by 0.7
Maryland            -2.8%      3.9%       D by 0.7
Michigan             6.9%      3.7%       R by 1.9#
North Carolina       7.8%      3.2%       R by 2.5#
Ohio                 6.8%      4.3%       R by 1.6
Pennsylvania         7.6%      3.2%       R by 2.4#
Texas                4.9%      3.1%       R by 1.6
Virginia             6.3%      3.4%       R by 1.9#
Wisconsin            7.0%      4.2%       R by 1.7#

                       Test 3 (Skewed Districts)

                      Imputing Uncontested Races

                    Minus                 [DELTA]
                    Median             (Average-Median
                  Democratic             Divided By
                   Vote (%)    Sigma       Sigma)

Arizona             -3.3%      3.8%       D by 0.9
Florida              4.8%      2.4%       R by 2.0#
North Carolina
Ohio                 6.8%      3.0%       R by 2.3#
Texas                7.0%      2.4%       R by 2.9#

In all cases, the last column gives the difference between
expectations and actual results, expressed in units of sigma, the
standard deviation, to give a measure that is comparable across the
three tests. Test 3 starts from raw percentage results and the last
column assumes voters in uncontested races are distributed 75%-25%
for the winning party. The boldface underlined entries indicate
statistically significant results. Test 2 could not be done for
Maryland because the grouped t-test requires each group to include
at least two wins.

Note: Statistically significant results are indicated with #.
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Title Annotation:II. Quantitatively Analyzing the Effects and Intents of Partisan Gerrymandering A. Analysis of Effects 3. National Districting Patterns Can Be Used to Identify a Natural Seats/Votes Relationship through Conclusion, with footnotes and tables, p. 1289-1321
Author:Wang, Samuel S.-H.
Publication:Stanford Law Review
Date:Jun 1, 2016
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