Gallagher Disproportionality Index
Also known as: Gallagher Index, Least Squares Index, LSq Index, Electoral Disproportionality Index
The Gallagher disproportionality index, also called the least squares index (LSq), is the standard summary measure of how faithfully an electoral system translates votes into seats. Introduced by Michael Gallagher in 1991, it takes the difference between each party's vote share and its seat share, squares those differences, sums and halves them, and takes the square root. Because deviations are squared before aggregation, the index gives disproportionate weight to a few large discrepancies rather than many small ones, capturing the intuition that one badly over- or under-represented party distorts the result more than scattered rounding errors. It has become the most widely reported single-number diagnostic of electoral-system performance in comparative political economy.
Key highlights
- Collapses an entire election's seat-vote distortion into one interpretable number on a percentage-point scale, enabling clean cross-national and over-time comparison.
- The least-squares formulation appropriately weights a few large deviations more heavily than many trivial ones, matching the substantive meaning of disproportionality.
- Requires only party vote shares and seat shares, data that are publicly available for virtually every democratic election.
- Has become a disciplinary standard, so reported values are directly comparable across the large literature that uses it.
Intuition
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How it works
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When to use it
Use the Gallagher index whenever you need a single, comparable number for how proportionally an election converted votes into seats — for ranking electoral systems, tracking a country's disproportionality over successive elections, or as a dependent or explanatory variable in cross-national models of party systems, representation, and government formation. It is the field-standard choice precisely because its squaring rule matches the substantive concern that a few large distortions matter most. It is less suitable when you specifically want every percentage point of distortion weighted equally (the Loosemore-Hanby index is then more appropriate), when the party-level vote and seat data are too coarse or inconsistently bounded to compute reliable shares, or when the analytic question is about the number or fragmentation of parties rather than the fidelity of the seat-vote translation, where the effective number of parties is the right tool.
Strengths & limitations
- Collapses an entire election's seat-vote distortion into one interpretable number on a percentage-point scale, enabling clean cross-national and over-time comparison.
- The least-squares formulation appropriately weights a few large deviations more heavily than many trivial ones, matching the substantive meaning of disproportionality.
- Requires only party vote shares and seat shares, data that are publicly available for virtually every democratic election.
- Has become a disciplinary standard, so reported values are directly comparable across the large literature that uses it.
- The result is sensitive to how small parties and the 'others' residual are treated, since bundling or splitting them changes the squared deviations.
- It summarizes the magnitude of distortion but not its direction or which type of party (large versus small) is favored.
- As a static, single-election measure it ignores how disproportionality interacts with strategic voting and party-system adaptation over time.
- Squaring makes the index dominated by the largest party's deviation, which can mask a pattern of systematic bias against many small parties.
Common pitfalls
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Applications
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Frequently asked
How does the Gallagher index differ from the Loosemore-Hanby index?
Both measure how far seat shares depart from vote shares, but they aggregate differently. The Loosemore-Hanby index sums the absolute values of the deviations and halves them, weighting every percentage point of distortion equally. The Gallagher least squares index squares the deviations before summing, halving, and square-rooting, so it gives more weight to a few large discrepancies than to many small ones. Gallagher argued this better matches the intuitive meaning of disproportionality, which is why his index has become the standard while Loosemore-Hanby is reported less often.
What counts as a high or low Gallagher value?
The index starts at zero for perfect proportionality and has no fixed upper bound, but in practice values cluster within recognizable ranges. Highly proportional list-PR systems typically score between about 1 and 5, mixed and moderately proportional systems fall in the middle, and majoritarian single-member-district systems such as the United Kingdom often produce values above 10 and sometimes above 15. These are empirical benchmarks rather than thresholds, and they depend on party-system fragmentation as well as electoral rules.
How are small parties and the 'others' category handled?
Treatment of minor parties materially affects the index because each contributes a squared deviation. Best practice is to include every party for which both a vote share and a seat share can be measured and to place only the unmeasurable residual into an 'others' bin, typically when its seat share is zero. Lumping many small parties together smooths the squared deviations and tends to understate disproportionality, so analysts should document their coding and, where possible, keep parties disaggregated for comparability with published series.
Sources
- 1.Gallagher, M. (1991). Proportionality, Disproportionality and Electoral Systems. Electoral Studies, 10(1), 33-51.
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ScholarGate. (2026, June 22). Gallagher Disproportionality Index. ScholarGate. https://scholargate.app/political-economy/gallagher-disproportionality-index