Gallagher index

The Gallagher index measures an electoral system's relative disproportionality between votes received and seats in a legislature.^[1][2] As such, it measures the difference between the percentage of votes each party gets and the percentage of seats each party gets in the resulting legislature, and it also measures this disproportionality from all parties collectively in any one given election. That collective disproportionality from the election is given a precise score, which can then be used in comparing various levels of proportionality among various elections from various electoral systems.^[3] The Gallagher index is a statistical analysis methodology utilised within political science, notably the branch of psephology.

Michael Gallagher, who created the index, referred to it as a "least squares index", inspired by the sum of squares of residuals used in the method of least squares. The index is therefore commonly abbreviated as "LSq" even though the measured allocation is not necessarily a least squares fit. The Gallagher index is computed by taking the square root of half the sum of the squares of the difference between percent of votes (${\displaystyle V_{i}}$) and percent of seats (${\displaystyle S_{i}}$) for each of the political parties (${\displaystyle i=1,\ldots ,n}$).^[4]

${\displaystyle \mathrm {LSq} ={\sqrt {{\frac {1}{2}}\sum _{i=1}^{n}(V_{i}-S_{i})^{2}}}}$  ^[5]

The division by 2 gives an index whose values range between 0 and 100. The larger the differences between the percentage of the votes and the percentage of seats summed over all parties, the larger the Gallagher index. The larger the index value the larger the disproportionality and vice versa. Michael Gallagher included "other" parties as a whole category, and Arend Lijphart modified it, excluding those parties. Compared to the Loosemore–Hanby index, the Gallagher index is more sensitive to large discrepancies.^[6] Other indices measuring the proportionality between seat share and party vote share are the Loosemore–Hanby index, Rae index, and the Sainte-Laguë Index.