Despite recent setbacks, litigation-based challenges to extreme partisan gerrymandering continue, and quantitative methods for detection are more important than ever. Many measurement tools have emerged that probe the question of whether a redistricting map is extreme or violates the principle of partisan symmetry. Such tools were used successfully in a lawsuit concerning Pennsylvania congressional districts under that state's constitution. Here we provide a framework for categorizing these tests for future use in state and federal constitutional cases. Our framework explains how measures should be interpreted and identifies which tests will be most effective given the specific facts of a particular state. Broadly, the tests can be divided into two categories: those that identify inequality of opportunity, i.e., a systematic effort to deprive of a group's ability to elect representatives; and those that identify inequality of outcome, i.e., a durable distortion in the amount of representation. In each case, the measures examine the difference between the existing map and what would occur under a districting process in which partisan interests are not the overriding consideration. A general thread is that of "significance testing," in which a district or statewide districting scheme can be defined as more extreme than the great majority of possibilities that could arise through unbiased means. Such tests are most often done with well-established classical statistical tests but can also include recently developed measures such as the efficiency gap. It is even now possible to evaluate, with mathematical rigor, whether a specific scheme is extreme relative to the virtually uncountable universe of possible maps. Taken together, these methods for detecting extremes comprise a statistical toolbox to address a wide variety of circumstances that may arise in any of the 50 states.
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