# Gallagher index

The Gallagher index is a statistical analysis methodology utilised within political science (notably the branch of psephology) which "measures an electoral system’s relative disproportionality between votes received and seats allotted 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]

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. Unlike the well-known Loosemore–Hanby index, the Gallagher index is less sensitive to small discrepancies.[6]

The Gallagher index gained considerable attention in Canada in December 2016 in the context of efforts to reform Canada's electoral system.[7][8] The Special Committee on Electoral Reform (a Parliamentary Committee) recommended "that the Government should, as it develops a new electoral system, use the Gallagher index in order to minimize the level of distortion between the popular will of the electorate and the resultant seat allocations in Parliament." The committee recommended that "the government should seek to design a system that achieves a Gallagher score of 5 or less."[9][10] In the 2015 Canadian federal election, the Gallagher index was 12.02, where 0 would be a perfectly proportional election outcome.[11]

## Examples of calculating disproportionality

#### New Zealand

This table uses the New Zealand 2005 election result.[12] 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.[13]

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 have historically been won by either the Māori Party, the Labour Party, or New Zealand First.

#### Australia

This table uses for example the 2012 Queensland state election, one of the largest landslides in Australian electoral history. Though Australia and New Zealand have somewhat similar political histories, Australia uses preferential voting in Single-member districts for Commonwealth House of Representative and most state and territory Legislative Assembly elections, which tends to result in far less proportionality compared to New Zealand's MMP system (or other proportional electoral systems), especially for larger minor parties, such as The Greens or, historically, the Australian Democrats. The 2012 Queensland election had an extremely high Gallagher Index, at 31.16, due to the massive landslide in seats for the victorious LNP. Most recent Australian state and federal elections however score between 10 and 12.

For comparison, the 2010 Australian Federal Election was more typical of Australian state and federal politics:

#### Sweden

The disproportionality of the 2018 Swedish general election was 0.63 according to the Gallagher index, which is extremely low by international standards (resulting in almost perfectly proportional seat allocations), due to Sweden's use of the modified Sainte-Laguë method in elections to the Riksdag.

#### Republic of Ireland

The disproportionality of the 2020 Irish general election was 1.96 according to the Gallagher index. The Republic of Ireland uses the single transferable vote (STV) system with Droop quota in elections to the Dáil Éireann.

#### United States

This table uses the aggregate results of the 2016 elections to the United States House of Representatives. These 435 single-seat elections are winner-take-all, which would tend to create disproportionate results, but this is moderated by the extremely high share of votes obtained by the two major parties—more than 97%, likely in part caused by fears of wasted votes and vote splitting.

party  % of votes  % of seats difference difference
squared
Republican Party 49.11% 55.40% -6.29 39.5915
Democratic Party 48.02% 44.60% 3.43 11.7390
Libertarian Party 1.29% 0% 1.29 1.6677
Independents and minor parties 1.18% 0% 1.18 1.4042
Green Party 0.39% 0% 0.39 0.1518
total of squares of differences 54.5541
total / 2 27.2771
square root of (total / 2) 5.22

#### Brazil

This table uses the results of the 2018 Brazilian general election (Lower House). The 513 seats are contested in statewide districts via open list proportional representation.

 Party % of votes Seats won % of seats Difference Difference² PSL 11.65 52 10.14 1.51 2.295 PT 10.30 56 10.92 -0.62 0.383 PSDB 6.01 29 5.65 0.35 0.124 PSB 5.85 34 6.63 -0.78 0.611 MDB 5.57 37 7.21 -1.64 2.689 PRB 5.53 34 6.63 -1.10 1.203 PSD 5.48 32 6.24 -0.76 0.578 PL 5.31 33 6.43 -1.12 1.254 PP 5.08 30 5.85 -0.77 0.595 PDT 4.66 29 5.65 -0.99 0.989 DEM 4.62 28 5.46 -0.84 0.698 PSOL 2.83 10 1.95 0.88 0.777 NOVO 2.79 8 1.56 1.24 1.525 PODE 2.28 11 2.14 0.14 0.019 PROS 2.08 8 1.56 0.52 0.268 PTB 2.06 10 1.95 0.11 0.012 SD 1.99 13 2.53 -0.55 0.300 AVANTE 1.88 7 1.36 0.51 0.261 PSC 1.80 8 1.56 0.24 0.056 PV 1.62 4 0.78 0.84 0.704 PPS 1.62 8 1.56 0.06 0.003 PATRI 1.46 5 0.97 0.48 0.232 PHS 1.45 6 1.17 0.28 0.079 PCdoB 1.35 9 1.75 -0.40 0.162 PRP 0.87 4 0.78 0.09 0.007 REDE 0.83 1 0.19 0.64 0.404 PRTB 0.70 0 0.00 0.70 0.485 PTC 0.64 3 0.58 0.06 0.004 PPL 0.61 2 0.39 0.22 0.049 DC 0.39 1 0.19 0.20 0.039 PMN 0.38 1 0.19 0.18 0.033 PMB 0.23 0 0.00 0.23 0.054 PCB 0.06 0 0.00 0.06 0.004 PSTU 0.04 0 0.00 0.04 0.002 PCO 0.00 0 0.00 0.00 0.000 100 531 100 Total 16.898 Total/2 8.449 Gallagher Index 2.91

## Other indices

Gallagher stated that the index constructed from and minimized by the Sainte-Laguë method "at the theoretical level is probably the soundest of all the measures."[14] 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. Special Committee on Electoral Reform (a Canadian Parliamentary Committee) (December 1, 2016). Report 3: Strengthening Democracy in Canada : Principles, Process and Public Engagement for Electoral Reform (Report). Parliament of Canada. p. 69 (or p. 83 in PDF search). Retrieved December 26, 2016. One tool that has been developed to measure an electoral system's relative disproportionality between votes received and seats allotted in a legislature is the Gallagher Index, which was developed by Michael Gallagher (who appeared before the Committee).
2. O'Malley, Kady (December 1, 2016). "Read the full electoral reform committee report, plus Liberal and NDP/Green opinions". Ottawa Citizen. Ottawa Citizen. Retrieved December 26, 2016.
3. This is discussed in simple English at "Gallagher Index Made Easy". 2016-12-31.
4. Gallagher, Michael (1991). "Proportionality, Disproportionality and Electoral Systems". Electoral Studies. 10: 33–51. doi:10.1016/0261-3794(91)90004-c.
5. Gallagher 1991, p. 40. sfn error: multiple targets (2×): CITEREFGallagher1991 (help)
6. Gallagher 1991, p. 41. sfn error: multiple targets (2×): CITEREFGallagher1991 (help)
7. Cash, Colby (December 2, 2016). "Colby Cosh: Did Maryam Monsef actually read the whole electoral reform report?". National Post. Retrieved 10 December 2016.
8. Wherry, Aaron (December 1, 2016). "Minister 'disappointed' as electoral reform committee recommends referendum on proportional representation". CBC News. Retrieved 10 December 2016.
9. O'Malley, Kady (December 1, 2016). "Read the full electoral reform committee report, plus Liberal and NDP/Green opinions". Ottawa Citizen. Ottawa Citizen. Retrieved 10 December 2016.
10. "Is Canada Fair?". Measuring Unfairness — Calculating Canada's Gallagher Index. (This website includes the Gallagher Index in adjustable table format. It initially shows the data for Canada's 2015 federal election, but some variables in some table cells are adjustable by the visitor to the website, and then the rest of the table is automatically adjusted to reflect this visitor's new input.). Retrieved 10 December 2016.
11. The rules for federal elections in Canada require that certain provinces always get a certain quantity of seats – on a province by province basis. If so, then Byron Weber Becker proposed that the Gallagher index for Canada ought to ALSO reflect that. In other words, the Gallagher data should be collected on a province by province basis; and the Gallagher score should be calculated on a province by province basis. Only after that is done, can we then add up all of those provincial scores and then average them out to get the true national "composite Gallagher index" score. If we do that, then the illustrated table calculation of 12 for Canada is incorrect. It should instead show a "composite Gallagher index" of 17.1. Byron Weber Becker developed this "composite" index. See citation here: Special Committee on Electoral Reform (a Canadian Parliamentary Committee) (December 1, 2016). Report 3: Strengthening Democracy in Canada : Principles, Process and Public Engagement for Electoral Reform (Report). Parliament of Canada. p. 69 (or p. 83 in PDF search). Retrieved December 26, 2016. ...Professor Becker developed the “Gallagher Index Composite” for the Committee’s study...
12. "Official Count Results – Electorate Status". Chief Electoral Office- New Zealand. 1 October 2005. Retrieved 9 August 2013.
13. Gallagher, Michael (2013). "Election indices" (PDF). Trinity College, Dublin. Retrieved 9 August 2013.
14. Gallagher 1991, p. 47. sfn error: multiple targets (2×): CITEREFGallagher1991 (help)