# Gallagher index

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One well-accepted measure of proportionality (for proportional representation) is the Gallagher index, which measures the difference between the percentage of votes each party gets and the percentage of seats each party gets in the resulting legislature, and aggregates across all parties to give a total measure in any one given election result. This measure attributes a specific level to a given election which can then be used in comparing various levels of proportionality among various elections from various Voting systems.

Michael Gallagher, who created the index, referred to it as a "least squares index", inspired by the residual sum of squares 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 (${\displaystyle i=1,\ldots,n}$).

${\displaystyle \mathrm{LSq} = \sqrt{ \frac{1}{2} \sum_{i=1}^n ( V_i-S_i ) ^2}}$

The index weighs the deviations by their own value, creating a responsive index, ranging from 0 to 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. Unlike the well-known Loosemore–Hanby index, the Gallagher index is less sensitive to small discrepancies.

While the Gallagher index is considered the standard measure for Proportional Representation, Gallagher himself considered the Sainte-Laguë index, which the Sainte-Laguë method optimizes, "probably the soundest of all the measures." This is closely related to Pearson's chi-squared test which has better statistical underpinning.

${\displaystyle \mathrm{SLI} = \sum {(S-V)^2 \over V}}$