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Voting matters - Issue 10, March 1999

Quantifying Representativity

P Kestelman

Philip Kestelman is keen on measuring electoral representativity, and works in family planning.

What is a Proportional Representation (PR) electoral system? Seriously begging that question, Gallagher (1991: 49) argued that "Each method of PR minimizes disproportionality according to the way it defines disproportionality, and thus each in effect generates its own measure of disproportionality".

However, Gallagher overlooked Single Transferable Voting (STV); an omission repaired by Hill (1997), invoking a 'Droop proportionality criterion' (DPC: Woodall, 1994: 10): "If, for some whole numbers k and m (where k is greater than 0 and m is greater than or equal to k), more than k Droop quotas of voters put the same m candidates (not necessarily in the same order) as their top m preferences, then at least k of those m candidates will be elected. In particular this must lead to proportionality by party (except for one Droop quota necessarily unrepresented) if voters decide to vote solely by party".

Thus defined, PR systems include Alternative Voting (AV: k=1); though over half the voters may be unrepresented! According to the 1937 Irish Constitution, not only parliamentary deputies (multi-member STV), but also the President (AV), shall be elected "on the system of proportional representation by means of the single transferable vote".

Yet nobody regards AV as a PR electoral system. In fairness to Woodall (1994: 10), "Any system that satisfies DPC deserves... to be regarded as a system of proportional representation (within each constituency)". At that level, Hill's "exaggerated case" (three-member STV) is persuasive; however disproportional to Party First Preferences. Nonetheless, constituency level 'PR' (including AV) is not enough for PR as normally construed.

(Hill (1992) reasoned that, if voters vote solely by party, each nominating sufficient candidates, "then STV will produce splendid proportionality, ... , while any discrepancy due to fractions of quotas can be expected to even out over a number of multi-member constituencies". Indeed, the main political question is how faithfully total seats reflect Party First Preferences overall (regionally and/or nationally).

Party Total Representativity

In parliamentary elections, the simplest measure of total disproportionality is the overall deviation between over-represented Party Seat-fractions and Vote-fractions: the Loosemore-Hanby Index (LHI),

LHI complements the Rose Index of Proportionality (RIP); for which I prefer the more explicit term, Party Total Representativity (PTR).

Table 1 below demonstrates the calculation of PTR = 100%-LHI for the 1997 Irish General Election, which proved unprecedentedly disproportional to Party First Preferences.

Table 1: STV Party (First Preference) Votes and Seats: Numbers, Fractions and Deviations: General Election, Irish Republic, 1997

* Loosemore-Hanby Index (LHI) = 12.9 percent = Overall deviation between over-represented Party Seat-fractions and Vote-fractions: complementing Party Total Representativity (PTR) = 87.1 percent. Source: Dáil Éireann (1998).

The Independent Commission on the Voting System (Jenkins, 1998: 47) gave a 1997 Irish General Election LHI of only 9.8 percent (their DV or 'deviation from pro-portionality': Dunleavy et al, 1997: 10). However, the two main parties (Fianna Fáil and Fine Gael) alone received 11.6 percent more Seats than Votes (First Preferences); and exact LHI=12.9 per cent (Table 1). LHI (and hence PTR) are often miscalculated.

Other Measures

McBride (1997: 9) invoked "O'Leary's index of proportionality": the ratio of each party's Seat-fraction to its First Preference Vote-fraction (S%/V%). However, the problem is how to combine such party-specific ratios (or deviations, S% - V%: see Table 1 above) into some measure of overall disproportionality. O'Leary (1979: 100) favoured the Rae Index of Disproportionality (RID), measuring party average disproportionality (contrast LHI above):

where N = Number of parties exceeding 0.5 percent of votes.

The palpable arbitrariness of this average disproportionality per party (why not a cutoff-point of 0.1 percent, or 5.0 per-cent of votes, for that matter?) may be redeemed somewhat by defining N as the 'effective number of parties' (Laakso and Taagepera, 1979):

Taagepera and Shugart (1989: 260) preferred N2 on practical grounds; though (entropy-based) N1 enjoyed "equally good conceptual credentials".

Gallagher (1991) argued that RID was "too sensitive to the number of parties"; to which LHI was "much too insensitive". Accordingly, he proposed a "least squares index": the Gallagher Index of Disproportionality,

Nevertheless, Gallagher (1991: 47) considered "probably the soundest of all the measures" the Sainte-Laguë Index,

Unfortunately, SLI ranges theoretically from zero to infinity; which Gallagher acknowledged was "less easily interpreted" than LHI or GID (ranging 0 - 100 percent). Thus in the 1997 Irish Presidential Election, AV First Count LHI = 55 percent (complementing PTR = 45 percent: President McAleese's First Preference Vote-fraction: Table 2 below); whereas SLI = 121 percent!

Table 2: AV Party Vote-fractions, Seat-fractions and Deviations, by Count: Presidential Election, Irish Republic, 1997

* First Count LHI = 54.8 percent: PTR = 45.2 percent.

** Final Count LHI = 44.4 percent: PTR = 55.6 percent.

Source: Irish Times, 1 November 1997.

Lijphart (1994: 60) preferred GID as steering "A middle course between the Rae and Loosemore-Hanby indices. Its key feature is that it registers a few large deviations much more strongly than a lot of small ones"; and contrasted two hypothetical elections (abstracted in Table 3 below).

Without defining any 'Lijphart Proportionality Criterion', he maintained that Election 1 was "highly disproportional" (GID = LHI = 5.0 percent); whereas Election 2 was "highly proportional" (GID = 2.2 percent; but LHI = 5.0 percent). Ironically, his intuitively "much more proportional" Election 2 yielded the higher SLI, considered by Gallagher (1991: 49) "the standard measure of disproportionality" !

Woodall (1986: 45) preferred the Farina Index,

FI is the angle between two multi-dimensional vectors, whose coordinates are Party Seat-fractions and Vote-fractions: theoretically ranging between 90° (cos FI=0) and zero degrees (parallel vectors: exact PR). As a fraction of a right angle, FID = FI/90°; so ranging 0 - 100 percent (instead of 0 - 90°).

In Table 3, FID (like RID and GID) evaluates Election 1 as more disproportional than Election 2. However, as Hill (1997) recognised, FID also poses problems of interpretation; remaining a far cry from the pristine simplicity of LHI.

Hill (1997) reproached PTR and other measures (their "fatal flaw") as confined to Party First Preferences. Nonetheless, he acknowledged that the concept of Total Representativity may be generalised (e.g. from Party to 'Cumbency', Gender and Name: Kestelman, 1996); and extended beyond the STV First Count. Yet he regarded Final Count PTR as merely comparing STV with itself!

Table 3: Five Measures of Overall Disproportionality: Two Hypothetical Elections

*As defined in the text above.

Minor / Micro-Parties

As Hill (1997) implied, minor parties and independents ('microparties' - representing nobody but themselves) may need disaggregating before calculating overall measures of disproportionality. Exact LHI necessitates disaggregating the votes for every represented party (and elected independent) from unrepresented parties; as in Table 1 above. SLI may also be calculated without disaggregating unrepresented parties.

On the other hand, exact GID requires disaggregating even unrepresented party votes. Moreover, in evaluating a few large deviations (S% - V%) as more disproportional than many small deviations, with the same total deviation (and hence LHI), GID implies that, the more fissiparously people vote, the more they deserve to be under-represented. In contrast, LHI consistently measures the total under-representation of all under-represented voters.


Gallagher (1991: 33-34) lamented that "There is remarkably little discussion of what exactly we mean by proportionality and how we should measure it ... how do we decide which is closer to perfect proportionality?" - when comparing different elections. Notice already two different senses of the term 'proportionality' here! Hence my preference for the term 'representativity' for measures admitting matters of degree to the relationship between votes and seats.

Gallagher (1991: 46) reported that, at 82 national elections in 23 countries (1979-89), LHI, GID and SLI (but not RID) proved impressively correlated with each other: so why complicate matters? Besides, measuring average disproportionality (RID) necessitates counting parties - a rather moveable feast - and there seems little virtue in quantifying some hybrid between the distinct concepts of total and average disproportionality

There remains legitimate scope for debating the relative merits of STV first or final preference representativity, in national aggregate or constituency average, respecting party or other considerations. In evaluating the representativity mediated by different electoral systems, no measure is perfect.

A generation after its introduction (Loosemore and Hanby, 1971), LHI survives relatively unscathed. I remain peculiarly susceptible to the complement (PTR) of that simplest LHI; doubting whether more complex measures of overall disproportionality would materially affect electoral comparisons (for example, STV representativity by District Magnitude: Kestelman, 1996).


  1. Dáil Éireann (1998): 28th Dáil General election, June 1997. Election Results and Transfer of Votes. The Stationery Office, Dublin.
  2. Dunleavy, P et al (1997): Making Votes Count. Replaying the 1990s General Elections under alternative electoral systems. Democratic Audit, Colchester.
  3. Gallagher, M (1991): 'Proportionality, Disproportionality and Electoral Systems': Electoral Studies 10, 33-51.
  4. Hill, I D (1992): 'STV and Party Proportionality: Another View': Representation 31: 113, 19.
  5. Hill, I D (1997): 'Measuring proportionality': Voting matters 8, 7-8.
  6. Jenkins, R (1998): The Report of the Independent Commission on the Voting System. Cm 4090-I. The Stationery Office, London.
  7. Kestelman, P (1996): 'Is STV a form of PR?': Voting matters 6, 5-9.
  8. Laakso, M and Taagepera, R (1979): '"Effective" Number of Parties: A Measure with Application to West Europe': Comparative Political Studies 12, 3-27.
  9. Lijphart, A (1994): Electoral Systems and Party Systems. Oxford University Press.
  10. Loosemore, J and Hanby, V (1971): 'The Theoretical Limits of Maximum Distortion: Some Analytic Expressions for Electoral Systems': British Journal of Political Science 1, 467-477.
  11. McBride, J P (1997): 'Positive Aspects of PR-STV for the Irish Political System': Representation 34:1, 7-12.
  12. O'Leary, C (1979): Irish Elections, 1918-1977: Parties, Voters and Proportional Representation. Gill and Macmillan, Dublin.
  13. Taagepera, R and Shugart, M S (1989): Seats and Votes. Yale University Press, New Haven.
  14. Woodall, D R (1986): 'How Proportional is Proportional Representation?': Mathematical intelligencer 8:4, 36-46.
  15. Woodall, D R (1994): 'Properties of Preferential Election Rules': Voting matters 3, 8-15.

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