Least squares index
The least squares index (LSq) was devised in the summer of 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.
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 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.)
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 1,100 elections in over 100 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.)
|English||least squares index|
|Albanian||Indeksi i katroreve me te vegjel|
|Danish||mindste kvadraters indekset|
|Dutch, Afrikaans||de kleinste kwadraten index|
|Finnish||pienin neliosumman hakemisto|
|French||l'indice des moindres carrés|
|German||Index der kleinsten Quadrate|
|Bahasa Indonesia||indéks kuadrat-kuadrat yang kecil sekali|
|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.
Last updated 9 December, 2016 5:47 PM