# Gallagher index

The Gallagher index measures an electoral system's relative disproportionality between votes received and seats in a legislature. 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. The Gallagher index is a statistical analysis methodology utilised within political science, notably the branch of psephology.
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 ($V_{i}$) and percent of seats ($S_{i}$) for each of the political parties ($i=1,\ldots ,n$).
    $\mathrm {LSq} ={\sqrt {{\frac {1}{2}}\sum _{i=1}^{n}(V_{i}-S_{i})^{2}}}$
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. Compared to the Loosemore–Hanby index, the Gallagher index is more sensitive to large discrepancies. Other indices measuring the proportionality between seat share and party vote share are the Loosemore–Hanby index, Rae index, and the Sainte-Laguë Index.

## History

The first publication of the use of least squares in measuring the dis-proportionality of election outcomes was by Michael Gallagher in 1991 in which he writes: "These [election] indices were originally outlined in Markku Laakso and Rein Taagepera, ‘ “Effective” number of parties: a measure with application to west Europe’, Comparative Political Studies 12:1 (1979), pp. 3–27 (effective number of parties), and Michael Gallagher, ‘Proportionality, disproportionality and electoral systems’, Electoral Studies 10:1 (1991), pp. 33–51 (least squares index)."

## Application in Canada

The Gallagher index gained considerable attention in Canada in December 2016 in the context of efforts to reform Canada's electoral system. 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."

## Examples of calculating disproportionality
### Canada

In the 2015 Canadian federal election, the Gallagher index was 12.02, where 0 would be a perfectly proportional election outcome.
Gallagher Index for the 2015 Canadian federal election :
| Party | Party | Votes (%) | Seats (%) | Difference | Difference squared |
| | Liberal | 39.47% | 54.44% | 14.97 | 224.1009 |
| | Conservatives | 31.89% | 29.29% | -2.6 | 6.76 |
| | New Democratic | 19.71% | 13.02% | -6.69 | 44.7561 |
| | Bloc Québecois | 4.66% | 2.96% | -1.7 | 2.89 |
| | Green | 3.45% | 0.29% | -3.16 | 9.9856 |
| | Other | 0.82% | 0.00% | -0.82 | 0.6724 |
| Total of differences squared | Total of differences squared | Total of differences squared | Total of differences squared | Total of differences squared | 289.165 |
| Total / 2 | Total / 2 | Total / 2 | Total / 2 | Total / 2 | 144.5825 |
| Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | 12.02 |

### 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. The LNP gained 88% of the seats with less than 50% of the vote. Most recent Australian state and federal elections however score between 10 and 12.
Gallagher Index for the 2012 Queensland state election :
| Party | Party | Votes (%) | Seats (%) | Difference | Difference squared |
| | Liberal National | 49.65% | 87.64% | 37.99 | 1443.2401 |
| | Labor | 26.66% | 7.87% | -18.79 | 353.0641 |
| | Katter | 11.53% | 2.25% | -9.28 | 86.1184 |
| | Greens | 7.53% | 0.00% | -7.53 | 56.7009 |
| | Other | 1.47% | 0.00% | -1.47 | 2.1609 |
| | Independent | 3.16% | 2.25% | -0.91 | 0.8281 |
| Total of differences squared | Total of differences squared | Total of differences squared | Total of differences squared | Total of differences squared | 1942.1125 |
| Total / 2 | Total / 2 | Total / 2 | Total / 2 | Total / 2 | 971.0563 |
| Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | 31.16 |

### EU

The 7 political groups of the European Parliament instead of the 203 political parties allow a concise calculation of disproportionality between votes and seats. The Gallagher index for the European Parliament is 7.87.
Gallagher Index for the 2019 European Parliament election :
| Party | Party | Votes (%) | Seats (%) | Difference | Difference squared |
| | EPP | 20.80% | 24.23% | 3.43 | 11.7649 |
| | S&D | 17.88% | 20.51% | 2.63 | 6.9169 |
| | RE | 12.01% | 14.38% | 2.37 | 5.6169 |
| | G/EFA | 10.04% | 9.85% | -0.19 | 0.0361 |
| | ID | 10.59% | 9.72% | -0.87 | 0.7569 |
| | ECR | 7.17% | 8.26% | 1.09 | 1.1881 |
| | GUE/NGL | 5.16% | 5.46% | 0.3 | 0.09 |
| | NI | 6.52% | 7.59% | 1.07 | 1.1449 |
| | Wasted vote | 9.82% | 0.00% | -9.82 | 96.4324 |
| Total of differences squared | Total of differences squared | Total of differences squared | Total of differences squared | Total of differences squared | 123.9471 |
| Total / 2 | Total / 2 | Total / 2 | Total / 2 | Total / 2 | 61.9736 |
| Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | 7.87 |

### Sweden

The disproportionality of the 2022 Swedish general election was 0.64 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 2012 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 96%, likely in part caused by fears of wasted votes and vote splitting. The Gallagher index ignores the effect of the primaries on the proportionality.
Gallagher Index for the 2012 United States House of Representatives elections :
| Party | Party | Votes (%) | Seats (%) | Difference | Difference squared |
| | Republican Party | 47.66% | 53.80% | 6.14 | 37.6996 |
| | Democratic Party | 48.77% | 44.20% | -4.57 | 20.8849 |
| | Libertarian Party | 1.11% | 0.00% | -1.11 | 1.2321 |
| | Independents and minor parties | 1.01% | 0.00% | -1.01 | 1.0201 |
| | Green Party | 0.31% | 0.00% | -0.31 | 0.0961 |
| | Others | 1.13% | 0.00% | -1.13 | 1.2769 |
| Total of differences squared | Total of differences squared | Total of differences squared | Total of differences squared | Total of differences squared | 62.2097 |
| Total / 2 | Total / 2 | Total / 2 | Total / 2 | Total / 2 | 31.1049 |
| Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | 5.58 |

### United Kingdom

The 2024 general election in the United Kingdom was the most disproportional in modern British history. The Liberal Democrats recorded their best ever seat result (72), despite receiving only around half the votes they did in 2010, and fewer votes overall than Reform, although the party's seat share was again lower than its share of the vote. Advocacy group Make Votes Matter found that 58% of voters did not vote for their elected MP. Make Votes Matter spokesman Steve Gilmore, Electoral Reform Society chief Darren Hughes, Reform UK leader Nigel Farage and the Green Party of England and Wales co-leader Adrian Ramsay were among the figures that called for electoral reform in the wake of the election. The campaigners said it was the "most disproportionate election in [British] history".
Gallagher Index for the 2024 United Kingdom general election :
| Party | Party | Votes (%) | Seats (%) | Difference | Difference squared |
| | Labour | 33.70% | 63.23% | 29.53 | 872.0209 |
| | Conservative | 23.70% | 18.62% | -5.08 | 25.8064 |
| | Reform UK | 14.29% | 0.77% | -13.52 | 182.7904 |
| | Liberal Democrats | 12.22% | 11.08% | -1.14 | 1.2996 |
| | Green | 6.39% | 0.46% | -5.93 | 35.1649 |
| | Independents | 1.96% | 0.92% | -1.04 | 1.0816 |
| | Others | 7.74% | 4.92% | -2.82 | 7.9524 |
| Total of differences squared | Total of differences squared | Total of differences squared | Total of differences squared | Total of differences squared | 1126.1162 |
| Total / 2 | Total / 2 | Total / 2 | Total / 2 | Total / 2 | 563.0581 |
| Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | Square root of (Total / 2): Gallagher Index result | 23.73 |

## Countries

Gallagher indices for individual countries. Only the last available index for each country is shown. :
| Country | Year | Gallagher Index |
| Albania | 2021 | 4.57 |
| Algeria | 2021 | 21.20 |
| Andorra | 2023 | 16.47 |
| Angola | 2022 | 4.32 |
| Antigua and Barbuda | 2023 | 8.79 |
| Argentina | 2023 | 5.14 |
| Armenia | 2021 | 10.74 |
| Australia | 2022 | 16.72 |
| Austria | 2024 | 3.21 |
| Bahamas | 2021 | 25.01 |
| Barbados | 2022 | 28.92 |
| Belgium | 2024 | 3.83 |
| Belize | 2020 | 23.51 |
| Benin | 2023 | 9.08 |
| Bermuda | 2020 | 19.02 |
| Bhutan | 2023-24 | 24.70 |
| Bolivia | 2020 | 2.60 |
| Bosnia and Herzegovina | 2022 | 5.30 |
| Botswana | 2019 | 12.26 |
| Brazil | 2022 | 3.53 |
| Bulgaria | 2023 | 3.77 |
| Burkina Faso | 2020 | 8.16 |
| Cabo Verde | 2021 | 3.54 |
| Canada | 2021 | 13.39 |
| Chile | 2021 | 9.58 |
| Colombia | 2022 | 4.60 |
| Costa Rica | 2022 | 8.26 |
| Croatia | 2024 | 7.04 |
| Cyprus | 2021 | 6.44 |
| Czech Republic | 2021 | 10.34 |
| Denmark | 2022 | 1.13 |
| Dominica | 2022 | 6.55 |
| Dominican Republic | 2020 | 7.32 |
| El Salvador | 2024 | 15.02 |
| Estonia | 2023 | 4.66 |
| EU | 2019 | 7.87 |
| Faeroe Islands | 2019 | 1.66 |
| Fiji | 2022 | 4.75 |
| Finland | 2023 | 3.99 |
| France | 2024 | 5.67 |
| Gambia | 2022 | 7.93 |
| Georgia | 2020 | 8.71 |
| Germany | 2021 | 3.48 |
| Ghana | 2020 | 2.64 |
| Gibraltar | 2023 | 2.45 |
| Greece | 2023 | 8.97 |
| Greenland | 2021 | 2.75 |
| Grenada | 2022 | 7.97 |
| Guatemala | 2023 | 8.85 |
| Guinea | 2020 | 2.14 |
| Guinea-Bissau | 2023 | 10.81 |
| Guyana | 2020 | 0.52 |
| Honduras | 2021 | 6.41 |
| Hungary | 2022 | 11.76 |
| Iceland | 2021 | 3.96 |
| India | 2024 | 6.98 |
| Indonesia | 2024 | 4.86 |
| Ireland | 2020 | 2.22 |
| Israel | 2022 | 4.40 |
| Italy | 2022 | 12.37 |
| Jamaica | 2020 | 20.62 |
| Japan | 2021 | 11.43 |
| Kosovo | 2021 | 3.58 |
| Latvia | 2022 | 10.65 |
| Lesotho | 2022 | 6.03 |
| Liberia | 2023 | 10.13 |
| Liechtenstein | 2021 | 5.62 |
| Lithuania | 2020 | 9.49 |
| Luxembourg | 2023 | 5.96 |
| Malawi | 2019 | 8.56 |
| Malaysia | 2022 | 6.99 |
| Malta | 2022 | 2.24 |
| Mauritius | 2024 | 31.38 |
| Mexico | 2024 | 15.58 |
| Moldova | 2021 | 8.44 |
| Monaco | 2023 | 10.37 |
| Mongolia | 2024 | 12.84 |
| Montenegro | 2023 | 4.93 |
| Morocco | 2021 | 4.18 |
| Mozambique | 2019 | 2.45 |
| Namibia | 2019 | 0.98 |
| Nepal | 2022 | 6.02 |
| Netherlands | 2023 | 1.46 |
| New Zealand | 2023 | 2.63 |
| Niger | 2020-21 | 8.10 |
| North Cyprus | 2022 | 7.78 |
| North Macedonia | 2024 | 3.42 |
| Northern Ireland | 2022 | 7.80 |
| Norway | 2021 | 3.65 |
| Panama | 2024 | 9.77 |
| Paraguay | 2023 | 9.53 |
| Peru | 2021 | 14.02 |
| Poland | 2023 | 6.46 |
| Portugal | 2024 | 5.94 |
| Romania | 2020 | 6.14 |
| Saint Kitts and Nevis | 2022 | 15.97 |
| Saint Lucia | 2021 | 28.97 |
| Saint Vincent and the Grenadines | 2020 | 10.37 |
| San Marino | 2024 | 3.88 |
| Sao Tome and Principe | 2022 | 7.04 |
| Scotland | 2021 | 7.03 |
| Senegal | 2022 | 2.73 |
| Serbia | 2023 | 3.80 |
| Seychelles | 2020 | 15.33 |
| Sierra Leone | 2023 | 2.44 |
| Singapore | 2020 | 21.67 |
| Slovakia | 2023 | 7.44 |
| Slovenia | 2022 | 11.49 |
| South Africa | 2024 | 0.48 |
| South Korea | 2020 | 13.65 |
| Spain | 2023 | 5.67 |
| Sri Lanka | 2020 | 4.54 |
| Suriname | 2020 | 8.70 |
| Sweden | 2022 | 0.64 |
| Switzerland | 2023 | 3.60 |
| Taiwan | 2024 | 8.19 |
| Thailand | 2023 | 5.51 |
| Timor-Leste | 2023 | 6.06 |
| Trinidad and Tobago | 2020 | 3.55 |
| Turkey | 2023 | 3.61 |
| Uganda | 2021 | 14.90 |
| United Kingdom | 2024 | 23.64 |
| United States | 2022 | 1.789 |
| Uruguay | 2019 | 1.73 |
| Venezuela | 2019 | 18.40 |
| Wales | 2021 | 9.36 |
| Zambia | 2021 | 5.21 |