# Gallagher Index

The Gallagher Index (or least squares index) is used to measure the disproportionality of an electoral outcome; that is, the difference between the percentage of votes received, and the percentage of seats a party gets in the resulting legislature. This is especially useful for comparing proportionality across electoral systems. The index involves taking the square root of half the sum of the squares of the difference between percent of vote and percent of seats for each of the political parties.

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

The index weighs the deviations by their own value, creating a responsive index, ranging from 0 to 100. The lower the index value the lower the disproportionality and vice versa. Michael Gallagher, who created the index, 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.[2]

## Example of calculating disproportionality

This table uses the New Zealand 2005 election result.[3] Note that since New Zealand uses the MMP voting system, voters have two votes. This list uses the party vote, which determines the proportionality of the House; the electorate vote determines the local member.

party  % of votes  % of seats difference difference
squared
Labour 41.10 41.32 0.22 0.0484
National 39.10 39.67 0.57 0.3249
NZ First 5.72 5.79 0.07 0.0049
Greens 5.30 4.96 0.34 0.1156
Māori 2.12 3.30 1.18 1.3924
United Future 2.67 2.48 0.19 0.0361
ACT 1.51 1.65 0.14 0.0196
Progressives 1.16 0.82 0.34 0.1156
Destiny 0.62 0 0.62 0.3844
Legalise Cannabis 0.25 0 0.25 0.0625
Christian Heritage 0.12 0 0.12 0.0144
Alliance 0.07 0 0.07 0.0049
Family Rights 0.05 0 0.05 0.0025
Democrats 0.05 0 0.05 0.0025
Libertarianz 0.04 0 0.04 0.0016
Direct Democracy 0.03 0 0.04 0.0016
99MP 0.03 0 0.03 0.0009
OneNZ 0.02 0 0.02 0.0004
Republicans 0.02 0 0.02 0.0004
total of squares of differences 2.5336
total / 2 1.2668
square root of (total / 2) 1.13

Thus the disproportionality of the 2005 New Zealand election is 1.13, which is very low by international standards.[4]

Note that the Māori Party has the highest difference, which is significantly above the others. This is due to New Zealand's system of reserved seats for Māori. The Māori seats are allocated by votes on a separate electoral roll, and while any party can contest these seats, they are generally won by either the Māori Party, the Labour Party, or New Zealand First.

The 2012 Election for the Australian state of Queensland resulted in a Gallagher Index of 31.16 - the winning party gained over 87% of the seats with just less than 50% of the vote

## Other indices

The Sainte-Laguë method is considered by Gallagher to be probably the soundest of all the measures. This is closely related to the Pearson's chi-squared test which has better statistical underpinning.

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

## Notes

1. ^ Gallagher 1991, p. 40.
2. ^ Gallagher 1991, p. 41.
3. ^ "Official Count Results -- Electorate Status". Chief Electoral Office. 1 October 2005. Retrieved 9 August 2013.
4. ^ Gallagher, Michael (2013). "Election indices" (PDF). Trinity College, Dublin. Retrieved 9 August 2013.