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{{No footnotes|date=November 2010}}
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]].

: <math>\mathrm{LSq} = \sqrt{ \frac{1}{2} \sum_{i=1}^n ( V_i-S_i ) ^2}</math>{{sfn|Gallagher|1991|p=40}}<!--What are ''V''<sub>''i''</sub> and ''S''<sub>''i''</sub>?-->

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 (academic)|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.{{sfn|Gallagher|1991|p=41}}

== Example of calculating disproportionality ==

This table uses the [[New Zealand general election, 2005|New Zealand 2005 election]] result.<ref name="General election, 2005">{{cite web |title=Official Count Results -- Electorate Status |url= http://www.electionresults.govt.nz/electionresults_2005/electoratestatus.html |publisher=[[Chief Electoral Office (New Zealand){{!}}Chief Electoral Office]] |accessdate=9 August 2013 |date=1 October 2005}}</ref> Note that since New Zealand uses the [[Mixed-member proportional representation|MMP voting system]], [[Elections in New Zealand|voters have two votes]]. This list uses the ''party vote'', which determines the proportionality of the [[New Zealand House of Representatives|House]]; the ''electorate vote'' determines the local member.

{| class="wikitable"
! party
! % of votes
! % of seats
! difference
! difference<br>squared
|-
| [[New Zealand Labour Party|Labour]]
| align="right" | 41.10
| align="right" | 41.32
| align="right" | 0.22
| align="right" | 0.0484
|-
| [[New Zealand National Party|National]]
| align="right" | 39.10
| align="right" | 39.67
| align="right" | 0.57
| align="right" | 0.3249
|-
| [[New Zealand First|NZ First]]
| align="right" | 5.72
| align="right" | 5.79
| align="right" | 0.07
| align="right" | 0.0049
|-
| [[Green Party of Aotearoa New Zealand|Greens]]
| align="right" | 5.30
| align="right" | 4.96
| align="right" | 0.34
| align="right" | 0.1156
|-
| [[Māori Party|Māori]]
| align="right" | 2.12
| align="right" | 3.30
| align="right" | 1.18
| align="right" | 1.3924
|-
| [[United Future New Zealand|United Future]]
| align="right" | 2.67
| align="right" | 2.48
| align="right" | 0.19
| align="right" | 0.0361
|-
| [[ACT New Zealand|ACT]]
| align="right" | 1.51
| align="right" | 1.65
| align="right" | 0.14
| align="right" | 0.0196
|-
| [[New Zealand Progressive Party|Progressives]]
| align="right" | 1.16
| align="right" | 0.82
| align="right" | 0.34
| align="right" | 0.1156
|-
| [[Destiny New Zealand|Destiny]]
| align="right" | 0.62
| align="right" | 0
| align="right" | 0.62
| align="right" | 0.3844
|-
| [[Aotearoa Legalise Cannabis Party|Legalise Cannabis]]
| align="right" | 0.25
| align="right" | 0
| align="right" | 0.25
| align="right" | 0.0625
|-
| [[Christian Heritage New Zealand|Christian Heritage]]
| align="right" | 0.12
| align="right" | 0
| align="right" | 0.12
| align="right" | 0.0144
|-
| [[Alliance (New Zealand political party)|Alliance]]
| align="right" | 0.07
| align="right" | 0
| align="right" | 0.07
| align="right" | 0.0049
|-
| [[New Zealand Family Rights Protection Party|Family Rights]]
| align="right" | 0.05
| align="right" | 0
| align="right" | 0.05
| align="right" | 0.0025
|-
| [[New Zealand Democratic Party|Democrats]]
| align="right" | 0.05
| align="right" | 0
| align="right" | 0.05
| align="right" | 0.0025
|-
| [[Libertarianz]]
| align="right" | 0.04
| align="right" | 0
| align="right" | 0.04
| align="right" | 0.0016
|-
| [[Direct Democracy Party of New Zealand|Direct Democracy]]
| align="right" | 0.03
| align="right" | 0
| align="right" | 0.04
| align="right" | 0.0016
|-
| [[99 MP Party|99MP]]
| align="right" | 0.03
| align="right" | 0
| align="right" | 0.03
| align="right" | 0.0009
|-
| [[One New Zealand Party|OneNZ]]
| align="right" | 0.02
| align="right" | 0
| align="right" | 0.02
| align="right" | 0.0004
|-
| [[The Republic of New Zealand Party|Republicans]]
| align="right" | 0.02
| align="right" | 0
| align="right" | 0.02
| align="right" | 0.0004
|-
| align="right" colspan="4" | '''total of squares of differences'''
| align="right" | 2.5336
|-
| align="right" colspan="4" | '''total / 2'''
| align="right" | 1.2668
|-
| align="right" colspan="4" | '''square root of (total / 2)'''
| align="right" | '''1.13'''
|}

Thus the disproportionality of the 2005 New Zealand election is 1.13, which is very low by international standards.<ref>{{cite web |last=Gallagher |first=Michael |title=Election indices |url= http://www.tcd.ie/Political_Science/staff/michael_gallagher/ElSystems/Docts/ElectionIndices.pdf |publisher=[[Trinity College, Dublin]] |accessdate=9 August 2013|authorlink=Michael Gallagher (academic) |format=PDF |year=2013}}</ref>

Note that the Māori Party has the highest difference, which is significantly above the others. This is due to New Zealand's system of [[Māori electorates|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.

== Other indices ==

The [[Sainte–Laguë Index]] 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.
:<math>\mathrm{SLI} = \sum {(S-V)^2 \over V}</math>

==Notes==
{{Portal|Politics}}
{{Reflist}}

== References ==
* {{cite journal |ref=harv |last1=Benoit |first1= Kenneth |year= 2000 |title=Which Electoral Formula Is the Most Proportional? A New Look with New Evidence |journal=Political Analysis |volume=8 |pages=381–388 |url= http://pan.oxfordjournals.org/content/8/4/381.short |accessdate=10 August 2013}}
* {{cite journal |ref=harv |last1= Gallagher |first1= Michael |authorlink=Michael Gallagher (academic) |year=1991 |title=Proportionality, Disproportionality and Electoral Systems |journal=Electoral Studies |volume= 10 |pages=33–51}}
* {{cite journal |ref=harv |last1= Gallagher |first1= Michael |authorlink=Michael Gallagher (academic) |year=1992 |title=Comparing Proportional Representation Electoral Systems: Quotas, Thresholds, Paradoxes and Majorities |journal=British Journal of Political Science |volume= 22 |pages=469–496}}
* {{cite book |ref=harv |editor-last1= Gallagher |editor-first1= Michael |editor1-link=Michael Gallagher (academic) |editor-last2=Mitchell |editor-first2=P |year=2005 |title=The Politics of Electoral Systems | location= [[Oxford]] | publisher= [[Oxford University Press]] |at= Appendix B |isbn =0-19-925756-6}}
* {{cite journal |ref=harv |last1= Kestelman |first1= Philip |date=March 1999 |title=Quantifying Representativity |journal=Voting matters |volume= 10 |url = http://www.votingmatters.org.uk/ISSUE10/P6.HTM |accessdate=10 August 2013}}

== External links ==
* [[:Image:Aus07Gallagher.PNG|Gallagher Index of the results]] of the [[Australian federal election, 2007|Australian 2007 federal election]]

[[Category:Voting systems]]
[[Category:Psephology]]
[[Category:Statistical terminology]]