We provide both a theoretical and empirical analysis of the Mean-Median Difference (MM) and Partisan Bias (PB), which are both symmetry metrics intended to detect gerrymandering. We consider vote-share, seat-share pairs $(V, S)$ for which one can construct election data having vote share $V$ and seat share $S$, and turnout is equal in each district. We calculate the range of values that MM and PB can achieve on that constructed election data. In the process, we find the range of vote-share, seat share pairs $(V, S)$ for which there is constructed election data with vote share $V$, seat share $S$, and $MM=0$, and see that the corresponding range for PB is the same set of $(V,S)$ pairs. We show how the set of such $(V,S)$ pairs allowing for $MM=0$ (and $PB=0$) changes when turnout in each district is allowed to be different. Although the set of $(V,S)$ pairs for which there is election data with $MM=0$ is the same as the set of $(V,S)$ pairs for which there is election data with $PB=0$, the range of possible values for MM and PB on a fixed $(V, S)$ is different. Additionally, for a fixed constructed election outcome, the values of the Mean-Median Difference and Partisan Bias can theoretically be as large as 0.5. We show empirically that these two metric values can differ by as much as 0.33 in US congressional map data. We use both neutral ensemble analysis and the short-burst method to show that neither the Mean-Median Difference nor the Partisan Bias can reliably detect when a districting map has an extreme number of districts won by a particular party. Finally, we give additional empirical and logical arguments in an attempt to explain why other metrics are better at detecting when a districting map has an extreme number of districts won by a particular party.

翻译：本文对旨在检测选区不公正划分的对称性指标——均值中位数差与党派偏差——进行了理论与实证分析。我们考虑得票率-席位率组合$(V, S)$，针对每个组合可构建满足得票率为$V$、席位率为$S$且各选区投票率相等的选举数据。我们计算了在此类构建数据上MM与PB所能取得的取值范围。在此过程中，我们确定了存在选举数据使得得票率为$V$、席位率为$S$且$MM=0$的$(V, S)$组合范围，并发现PB对应的取值范围与此相同。我们展示了当允许各选区投票率不同时，满足$MM=0$（及$PB=0$）的$(V,S)$组合集合如何变化。尽管存在选举数据使$MM=0$的$(V,S)$集合与使$PB=0$的集合相同，但MM与PB在固定$(V, S)$上的可能取值范围并不一致。此外，对于固定的构建选举结果，均值中位数差与党派偏差的理论最大值可达0.5。我们通过实证表明，在美国国会选区地图数据中这两个指标值的差异最大可达0.33。运用中性集合分析与短时爆发法，我们证明无论是均值中位数差还是党派偏差，都无法可靠检测选区划分地图是否出现特定政党赢得极端数量选区的情况。最后，我们通过补充实证与逻辑论证，尝试解释为何其他指标能更有效地检测选区划分地图中特定政党赢得极端数量选区的现象。