Research Report 4 - Australian Federal Redistributions 1901-2003: Appendix One

Updated: 30 May 2013

Appendix One: Measures of Malapportionment

David-Eisenberg Index

The David-Eisenberg Index is calcu1ated as the ratio of the largest electorate (in terms of enrolment size) to the smallest. Therefore, the greater the variation between the largest and smallest electorate, the higher would be the Index measure. The following calculation illustrates the David-Eisenberg Index:

Illustration of the David-Eisenberg Index
State Smallest Largest David-Eisenberg Index
  (a) (b) (b/a)
South Australia (1934) 48 511 61 563 1.2691

The closer the final calculation is to 1.0, the more equal is the variation. Obviously this index is not capable of use in a situation where there are only 2 electorates, as is currently the case in the Australian Capital Territory and the Northern Territory.

Dauer-Kelsay Index

The Dauer-Kelsay Index is the smallest percentage of the total enrolment contained in the electorates required to produce a majority in the legislature.

It is calculated by listing electorates in ascending size of enrolment, then going up the list until a majority of electorates has been taken and then calculating the enrolment totalled to that point as a percentage of the enrolment for the whole legislature.

The closer the index result is to 0.5 the more equal the calculation.

Gini Index

The Gini Index is a general measure of inequality in the social sciences. The closer the index is to 0.0 the more equal the calculation is.

This is best explained by reference to the following hypothetical example where the axes of the diagram are the number of seats (horizontal) and the enrolments (vertical). In the following example if voters were equally apportioned amongst the number of seats available 20 percent of voters would have been located in 20 percent of the seats, 40 percent in 40 percent seats and so on. Such a distribution is shown by the 'line of equality'.

However, in this example voters are not equally apportioned and the plotting of the enrolments and seats (beginning with the smallest and moving to the largest) produces a curve, the 'Lorenz Curve'. The curve meets the line of equality at the top right of the diagram where 100 percent of the enrolments are contained within all of the seats.

image shows the Lorenz curve and the Gini index

The Gini Index is calculated as the ratio of the shaded area between the line of equality and the Lorenz Curve to the hatched triangular area to the right of the line of equality. The scale of the Gini Index ranges from zero for an equal electoral distribution to unity (1.000) for a malapportioned distribution so extreme that all electors are located in only one of the electoral districts.

In 1983 the Joint Standing Committee Inquiry necessitated that the Committee assess the equality of Australia's electoral systems. Three measures of malapportionment were referred to in evidence. While the Committee made use of the David-Eisenberg Index and the Dauer-Kelsay Index, the Gini Index was more commonly referred to in submissions. The Committee also noted comments made previously by the then Electoral Commissioner, Professor Colin Hughes, that the Gini Index was regarded as a better measure of equality:

The curve and the (shaded) area between the curve and the line of equality are better measures of deviation from equality across the whole range of electorates than either (the David-Eisenberg Index or the Dauer-Kelsay Index) because the location of each electorate in relation to the line of equality is measured.[12]

12 Colin A. Hughes, A Handbook of Australian Politics and Government, (Canberra: ANU Press) 1977, p. 129.