The Gallagher Index (or least squares) is used to measure the disproportionality of electoral systems, that is the difference between the percentage of votes received and the percentage of seats a party gets in the resulting legislature, in particular for majoritarian 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 (as whole numbers) for each of the political parties.

$LSq\; =\; \backslash sqrt\{\; \backslash frac\{1\}\{2\}\backslash Sigma\; (\; V-S\; )\; ^2\}$

The index weighs the deviations by their own value, creating a responsive index. 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.

Example of calculating disproportionality

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This table is using the New Zealand 2005 election result (note: New Zealand 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	Percentage of votes	Percentage 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				2.4711
HALVED				1.23555
SQUARE ROOT				1.11

Thus the disproportionality of the 2005 New Zealand election is 1.11

Other indices

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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-square test which has better statistical underpinning.

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

See also

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References

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• Gallagher, M (1991): Proportionality, Disproportionality and Electoral Systems: Electoral Studies 10, 33-51.
• Gallagher, M and Mitchell P. (eds), The Politics of Electoral Systems Oxford University Press, 2005, Appendix B. ISBN 0-19-925756-6

External links

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• Voting matters - Issue 10

Voting systems | Psephology