Loosemore–Hanby index


The Loosemore–Hanby index measures disproportionality of electoral systems. It computes the absolute difference between votes cast and seats obtained using the formula:
where
is the vote share and
the seat share of party
such that,
and is the overall number of parties.
This index is minimized by the largest remainder method with the Hare quota. Any apportionment method that minimizes it will always apportion identically to LR-Hare. Other methods, including the widely used divisor methods such as the Webster/Sainte-Laguë method or the D'Hondt method minimize other disproportionality indexes instead.
The index is named after John Loosemore and Victor J. Hanby, who first published the formula in 1971 in a paper entitled "The Theoretical Limits of Maximum Distortion: Some Analytic Expressions for Electoral Systems". Along with Douglas W. Rae's, the formula is one of the two most cited disproportionality indices. Whereas the Rae index measures the average deviation, the Loosemore–Hanby index measures the total deviation. Michael Gallagher used least squares to develop the Gallagher index, which takes a middle ground between the Rae and Loosemore–Hanby indices.
The LH index is related to the Schutz index of inequality, which is defined as
where is the expected share of individual and her allocated share. Under the LH index, parties take the place of individuals, vote shares replace expectation shares, and seat shares allocation shares. The LH index is also related to the dissimilarity index of segregation. All three indexes are special cases of the more general index of dissimilarity.

Works cited