# Indices How the archive measures party systems: what each index means, how it is computed, and the judgment calls behind the numbers. ## Which elections get indices The archive computes indices for a chamber when the chamber is filled by party and the votes behind it were counted. That means at least 10 seats, results for at least half of them, party candidates in at least half of the seats won, and at least two parties with votes. Presidents, mayors, and other single-seat races get no indices, and neither do nonpartisan bodies. The figures use the chamber's general election as a whole. In a two-round system, votes come from the first round, when every party is on the ballot, and seats come from the final result. In a mixed system such as Germany's or Italy's, votes are the party list votes, and seats are the chamber's full membership across every tier. Under the single transferable vote, as in Ireland, votes are first preferences. ## Effective number of parties The effective number of parties, from Markku Laakso and Rein Taagepera (1979), counts parties by weighing each one by its size: `N = 1 / Σ pᵢ²`, where `pᵢ` is a party's share of the votes or of the seats. Ten equal parties give 10; one party with 70% and three with 10% each give about 1.9. The archive reports it twice: by votes, for the electorate's choices, and by seats, for the chamber that came out of them. Rae fragmentation, `F = 1 − Σ pᵢ²`, is the same sum read another way: the chance that two voters, or two members, picked at random belong to different parties. The API returns it beside the effective number of parties. ## Disproportionality The Gallagher index, or least squares index, from Michael Gallagher (1991), measures how far seats stray from votes: `LSq = √(½ Σ (vᵢ − sᵢ)²)`, with vote and seat shares in percent. Squaring the gaps makes one large gap count for more than many small ones, so a winner's bonus under first past the post shows up clearly. The United Kingdom's 2024 election scored 23.7. The Loosemore–Hanby index, `D = ½ Σ |vᵢ − sᵢ|`, adds up the gaps without squaring them. It reads as the share of seats that went to parties beyond their share of the vote. It runs higher than the Gallagher index when many small parties each fall a little short. ## Volatility Pedersen volatility, from Mogens Pedersen (1979), measures how much of the vote changed hands since the previous election: `V = ½ Σ |pᵢ,t − pᵢ,t−1|`. The previous election is the same chamber's last general election in the archive. A party's identity across two elections is the hard part. The archive joins two parties when its records say one merged into, succeeded, was refounded as, or was absorbed by the other, served as the other's electoral list, or is a regional branch of it. It follows the records up to four steps, and only as of the election date. A renamed party is the same party in the archive, so a new name never counts as change. A party that split off counts as a new party, so a split shows up as volatility. When a party with 10% of the vote or more has no match on the other side, the figure carries a note: the whole share counts as change, which may reflect a gap in the records more than it reflects voters moving. ## Nationalization The party system nationalization score, from Mark Jones and Scott Mainwaring (2003), measures how evenly parties' votes spread across the country. For each party, it takes the Gini coefficient of the party's vote share across the places where votes were counted. The formula is `PNS = 1 − Gini`. The score for the whole system weighs each party's score by its share of the vote: `PSNS = Σ pᵢ × PNSᵢ`. The archive uses it over other measures because it runs from 0 to 1, is simple to compute and explain, and is widely used. That makes it easy to compare with published work. The places are the finest level the archive reports for the chamber's counted votes, such as constituencies in the United Kingdom or states in Germany, and each counts once regardless of size. The Gini coefficient is sensitive to the number of places, so a score measured over fewer than ten carries a note. ## Judgment calls Each of these choices changes the numbers, so each figure lists the ones that applied to it. - **Independents and others.** Some sources report independents or minor candidates as one group. For the effective number of parties by votes, that group counts as many tiny parties, not one large one, and each independent seat counts as its own party. The disproportionality indices treat the group as one row. - **Alliances.** When a source reports an alliance's seats and credits them to its member parties, the archive credits them too. When an alliance's votes aren't split by party, as in Italy, the alliance counts as one party, in both elections when it measures volatility. - **Sister parties.** Parties in a standing alliance that never run against each other, such as Germany's CDU and CSU, count as one party. - **Uncontested seats.** Seats won unopposed have no votes, so they are left out of every figure, and the figure says how many. - **Rounds and tiers.** First-round votes in two-round systems, list votes in mixed systems, and first preferences under the single transferable vote, as described above. ## Getting the numbers Every figure comes with its inputs and notes from the API: `/api/v1/elections/{id}/indices` for each chamber an election filled, `/api/v1/contests/{id}/indices` for a list contest, and `/api/v1/polities/{id}/indices` for a country's whole series. The same data is in the MCP tools `get_election_indices`, `get_contest_indices`, and `get_polity_indices`, and in the Markdown and JSON versions of election and country pages. Also available as HTML at https://electionsarchive.com/indices and JSON at https://electionsarchive.com/indices.json.