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Least squares index


The least squares index (LSq) was devised in the late 1980s (17 July 1989) as a measure of the amount of disproportionality generated by an election outcome, by which is meant the disparity, if any, between the distribution of votes at the election and the allocation of seats. There are various ways in which this could be measured and they do not all produce identical rankings of a particular seat allocation; in other words, an allocation that minimises disproportionality (i.e. is the most proportional) according to one measure will not necessarily be the most proportional by another measure. The construction of the index owed much to the work of contemporary electoral systems scholars and also to an article published in 1911 by the French mathematician André Sainte-Laguë. The index was outlined in an article (download front page of article here; full article available from site of the journal Electoral Studies) published in 1991.

LSq formula


While measures of disproportionality rarely generate much public interest, the least squares index became the subject of controversy in the Canadian House of Commons on 1 December 2016 when Maryam Monsef, the Minister of Democratic Institutions in the Liberal Party government, expressed disappointment at what she saw as an unsatisfactory report from a parliamentary committee on election reform. She complained twice that whereas she had hoped for a recommendation for a specific alternative electoral system, 'Instead they've provided us with the Gallagher index', holding up a piece of paper with the least squares formula printed in large font upon it as she spoke. (See pictures here and here, the second being a story (seemingly no longer available online) from the Montréal newspaper La Presse (5 December 2016), where it is dubbed 'L'image du jour', and story and video clip on CBC site, plus interview with the National Post.) Canadian sales of The Politics of Electoral Systems boomed in the following days, or so its editors would like to think.




More information about the least squares index and about the effective number of parties, a measure of fragmentation, can be found on the accompanying site, which contains

(i) the values of the index for over 1,300 elections in over 130 countries;

(ii) a downloadable file explaining in detail how to calculate the value of the index for any given election outcome;

(iii) a downloadable file enabling users to calculate the value of the index themselves for any election outcome.

There is also much more information in the book The Politics of Electoral Systems, paperback edition (Oxford University Press, 2008).



(Unfortunately, some characters, eg those with an accent, an umlaut or equivalent, are impossible to reproduce accurately in this format; the attached file LSqTrans.pdf (40kb), viewable or downloadable here, shows how the translations should look.)

Language Translation
English least squares index
Albanian Indeksi i katroreve me te vegjel
Danish mindste kvadraters indekset
Dutch, Afrikaans de kleinste kwadraten index
Estonian vahimruudu naitaja
Finnish pienin neliosumman hakemisto
French l'indice des moindres carrés
German Index der kleinsten Quadrate
Bahasa Indonesia indéks kuadrat terkecil
Irish Innéacs na nÍoschearnóg
Italian l'indice di minimi quadrati
Norwegian minst rutene indeksen
Polish indeks najmniejszych kwadratów
Portuguese o mínimos quadrados índice
Spanish el índice de mínimos cuadrados
Swedish minsta kvadratmetoden index


Another concept widely used in the analysis of electoral systems is the effective threshold.

Some political photographs.



Cover of the book 'The Politics of Electoral Systems' Cover of the book 'The Politics of Electoral Systems' Cover of the book 'The Politics of Electoral Systems'


Last updated 20 August, 2021 11:57 AM