2020 Apportionment of Seats: Part 3 – Changing the Formula for States

(Update 3 July – the determination has been published confirming that Victoria will gain a seat and Western Australia and the Northern Territory lose seats. Details here.)

On 3 July, Australian Electoral Commissioner Tom Rogers will issue his determination on how many electorates will be contested and representatives elected for each state and territory at the next federal election. The determination will be based on Australian Bureau of Statistics population statistics to be released this week.

Based on population trends, it is expected that Victoria will gain a seat to 39 seats, and Western Australia will lose the 16th seat it gained in 2016. More controversially, the Northern Territory will lose the second member it has elected at every election since 2001.

This is the third of three posts on the subject of apportioning seats to states and territories under Australian constitutional and electoral law.

The first post looked at the constitutional allocation of seats to states under Section 24 of the Constitution, how the current formula works, past attempts to change the formula, and how past High Court cases have interpreted the workings of Section 24.

The second looked at the constitutional basis and history of territory representation. As I explain in the post, the allocation of seats to the territories is governed by legislation, not the constitution. The Parliament can change the territory allocation formula, and I propose that it should be changed to use what is known as Dean’s method. This would provide a fairer and more stable method of allocating seats than the current formula, though it would not guarantee the Northern Territory two seats into the future.

A private member’s bill has been introduced in the Senate to guarantee a minimum two seats for the Northern Territory. The Joint Standing Committee on Electoral Matters has launched an inquiry into the bill with submissions closing on 10 July. You can find details of the inquiry here.

In this post I will re-cap the US apportionment methods I discussed in my post on the territories and ask whether they could also be applied to the Australian states without risking the wrath of the High Court. In short my findings are that across 26 Australian apportionments since Federation, Dean’s method would have added one seat to one state at one of the 26 apportionments, one change out of 416 state allocations.

For this reason I argue that switching formula to adopt Dean’s method would meet the tests for changing the constitutional formula discussed in McKellar’s case (1977). (See me first post for details). It can be argued that Dean’s method, by minimising the difference between the average enrolment in each state and the national quota, provides a more proportional method than the variant of Webster’s method set out in Section 24 of the constitution.

Three American Methods – Webster, Huntington-Hill and Dean

First a quick recap of the US methods explained in my post on allocating seats to the territories.

The US Constitution states that representation in the House of Representatives is to be allocated to states according to their population, but every state is guaranteed one representative. The size of the House is determined by the Congress and there is no link between the size of the House and the Senate. Apportionments take place every ten years after a national Census.

Before going into details of the methods, there are a few details of the US apportionment process that do not apply or are of little relevance to apportionment in Australia.

  • At 50 there are many more states in the USA, and many more small states. Currently seven US states having a single member, five have two members. Four states have more than 25 members and the largest state California has 53.
  • The US House of Representatives apportionments have been fixed at 435 seats for the last century. To make the outcome match the size of the House, the process has to be applied in iterations, multiple passes adding or subtracting seats rather than the one pass method applied by Section 24. (The US formulas are now applied using divisors and allocation tables, a technique that is a little beyond the scope of this post.)
  • Two centuries of apportionments, changes in size of the House, accession of new states, and use of different methods, has created a vast test bed of data sets and different methods of allocation. Each subtle difference created by methods or data is closely examined to argue for one method or another.
  • The fixed size and different available methods mean there is more debate over gains or losses of seats by a state in relation to gains or losses by other states.
  • The USA has a minimum one seat for each of state and Australia has a minimum five seats for original states, which distorts many of the fairness measures used in the US literature on apportionment.

Every apportionment since 1940 has been undertaken using what is known as the Huntington-Hill method (or Huntington, or Hill, or the method of equal proportions). Webster’s method was used after the US Census of 1840, 1910 and 1930.

The method I am proposing be used in Australia, Dean’s method, has not been used for a US apportionments. However, it has been proposed for use, has in the past been used as the basis for Supreme Court apportionment cases, and its properties are discussed in the literature on US apportionment.

The complexity of apportionment is created by a state’s proportion of the national population being a real number (a number with decimals), but the number of seats to be allocated must be an integer (a whole number). A state might have population corresponding to 1.5 quotas of people, but it can only be allocated whole numbers of seats, in the case of 1.5 quotas, either one or two seats.

The Australian formula in Section 24 of the Constitution is Webster’s method, except the lack of a fixed size to the House means that seats are allocated by a simple formula rather than an iterative process required to produce a fixed size House.

All three methods start with the same step – calculate a quota by dividing the population by the number of seats to be allocated. In the case of the US, the current divisor is 435, in Australia it is currently 144 or twice the number of state senators. (I explained in the first post why the Australian formula includes only the state population and the state senators in the formula.)

All three methods then calculate a quotient for each state by dividing the population of the state by the national quota. All three methods then allocate a number of seats equal to the integer part of the quotient, that is the whole number to the left of the decimal place. For example, at the 2017 Australian apportionment, the quotient for Victoria calculated as 37.89234. All three methods allocated 37 seats to Victoria.

For the Australian territories, the same quota is also applied to the population of each territory with some minor tweaks as I explained in the second post.

Where the three allocation methods differ is in how they deal with the decimal part of the quotient, that is the numbers to the right of the decimal point. In the case of Victoria in 2017, the fraction was 0.89234. In this case all three methods would round up and allocate Victoria a 38th seat.

Each method uses a different rounding point to allocate seats to the upper or lower bound of seats. For Victoria, Webster’s method would round at 37.5000, Huntington-Hill at 37.4967 and Dean at 37.4933. Compared to Webster’s method, these represent differences in state population of 550 and 1,099. In other words, for large states, the chances of the three methods producing different outcomes is very small.

To explain the formula, I can’t avoid a little bit of mathematics. Let me define two variables –

  • L is the lower bound, the whole number below the quotient
  • U is the upper bound, the whole number above the quotient. The upper bound is always one seat higher than the lower bound, so it can be expressed as U = L + 1

The difference between the methods is created by the different rounding points each uses in allocating a state’s seats to the upper and lower bounds –

  • Webster’s Method – rounds at the arithmetic mean of the upper and lower bound, a value that always equals 0.5. In algebra it is  (L + 0.5), or (L + U) / 2, or (2L + 1) /2.
  • Huntington-Hill Method – rounds at the geometric mean of the upper and lower bound, that is  sqrt(L*U), or sqrt(L*(L+1))
  • Dean’s Method – rounds at the harmonic mean of the lower and upper bounds., that is 2*L*U / (L + U) or 2*L*(L+1)/(2L+1)

Each of these minimises different properties of fairness and proportionality –

  • Webster’s method minimises the absolute differences between states in the number of seats allocated for a given population.
  • Huntington-Hill minimises the relative differences in the people per member for each state.
  • Dean’s method minimises absolute differences in the number of people per member for each state.

In the academic literature it is argued that Dean’s method favours smaller states, Webster’s method favours larger states (this is disputed), and Huntington-Hill falls somewhere in between.

I think much of this debate is nullified in Australia by having a variable sized House, and by the minimum five seats per original state rule. The US debate is also coloured by state representation determining electoral college votes for electing the President, as well as the winner-takes-all method of allocating Electoral College votes to candidates. The Electoral College process has an in-built bias to smaller states due to the inclusion of Senate representation.

(If you are greatly interested in US apportionment and can cope with algebra, I would refer you to the Balinsky and Young reference at the end of this post.)

Table 1 below uses the above formulas to calculate rounding points for the three methods in electing between 1 and 12 members per state or territory.

Table 1 –  Comparing Rounding Points for the Methods

Webster Huntington-
1 2 1.5000 1.4142 1.3333
2 3 2.5000 2.4495 2.4000
3 4 3.5000 3.4641 3.4286
4 5 4.5000 4.4721 4.4444
5 6 5.5000 5.4772 5.4545
6 7 6.5000 6.4807 6.4615
7 8 7.5000 7.4833 7.4667
8 9 8.5000 8.4853 8.4706
9 10 9.5000 9.4868 9.4737
10 11 10.5000 10.4881 10.4762
11 12 11.5000 11.4891 11.4783

For very small numbers of seats allocated, the three methods can have a substantial impact. In my post on the territories, I highlighted the extra elections at which the NT and ACT would have been allocated an extra seat. (Check my post on territory representation for detailed discussion of the differences.)

However, the differences between the three methods disappear as the number of seats to be allocated increases. Would any past state apportionment have been changed by the different methods?

Would Alternative Methods have produced Different Outcomes at Past Australian State Apportionments?

To test how the Huntington-Hill and Dean methods could have altered past Australian state apportionments, I have calculated both methods for every state since the current determination method was adopted in 1984. I have also applied them to population estimates published by the former Commonwealth Electoral Officer as far back as the first apportionment review in 1904.

The impact of using the Huntington-Hill and Dean methods as opposed to the Webster (section 24) method can be summarised as follows –

  • There is no difference between the three methods for any state apportionment determination since the current method was adopted in 1984.
  • The Huntington-Hill method produced no difference compared to the current method for any state apportionment prior to 1984.
  • The Dean method’s harmonic mean would have changed one allocation in one state prior to 1984, and come close to changing one other apportionment.

In summary, for 26 national or 156 individual state apportionment determinations since 1904, the Huntington-Hill method produced no differences to the existing method set out in Section 24 of the constitution.

Dean’s method had a one seat change at one out of the 26 apportionments, or one out of 156 individual state apportionments.

(BUT – both methods would have altered the apportionment of seats to territories, as I explained in my second post.)

The one case where Dean’s method produced a different state allocation, and the one where it was nearly different, are discussed below.

The 1967 Apportionment

The table below shows the population figures published by the Commonwealth Electoral Officer in 1967 after the 1966 Census. I have left Tasmania out of the table for being guaranteed five seats, and the two territories, which at the time were granted one seat each without application of the formula. On a historical note, at this time South Australia still had a larger population than Western Australia.

The original population numbers did not include quotas or quotients. I have calculated both using the method used today by the AEC. On that basis the national quota was 95,352.

Table 2 shows the 1967 apportionment with the South Australian line highlighted to show the difference between the Webster and Dean methods.

Table 2 –  The 1967 Apportionment

Population Quotient Webster Dean
NSW 4,237,458 44.4400 44 44
VIC 3,217,855 33.7470 34 34
QLD 1,672,958 17.5450 18 18
SA 1,094,767 11.4813 11 12
WA 847,935 8.8926 9 9

The mathematics in Dean’s method hides the simple property by which it operates. Dean’s formula allocates seats equal to the lower or upper bound based on which would produce an average population per member that is closer to the national quota.

For South Australia, the average voters per member if 11 seats were allocated was 99,524, a difference of 4,172 from the national quota. If 12 seats were allocated, the average was 91,231, a difference of 4,121. Dean’s method rounds to the bound that produces an average closer to the national quota, which in 1967 would have resulted in South Australia being allocated a 12th seats, an increase since 1961.

If a proportional allocation is one that minimises the difference between a state’s average population per member and the national quota, than Dean’s method is more proportional than the Webster’s method set out in Section 24 of the Constitution.

On a historical note, Webster’s method was not applied for the 1967 apportionment. At the time, a formula inserted in the Representation Act had replaced the Section 24 method. Any state where the calculated quotient had a remainder was automatically rounded up.

Using the data in Table 2, the any-decimal method used in 1967 rounded South Australia’s 11.4813 up to 12 seats, and NSW’s 44.4400 was rounded up to 45 seats. This was the method of rounding the High Court invalidated in McKellar’s case in 1977.

In McKellar’s case the High Court ruled that the automatic rounding up rule produced an outcome that was both less proportional than the method in Section 24, and produced a House that was not “as near as practicable” to twice the size of the Senate. Several of the justices ruled entirely on the “practicable” test without considering proportionality. (Refer to my first post for a lengthy discussion of McKellar’s case.)

But as Justice Stephen noted in his reasoning,  proportionality is set out in Section 24  in absolute terms, but “as near as practicable” is in qualified terms. Given the formula is written as “until the parliament otherwise decides”, is Parliament free to legislative for a formula that is more proportional than the formula in Section 24, even if in certain circumstances it can produce an outcome that is less “near as practicable”?

In my second post I set out why I believe that Dean’s method should be used in allocating seat to the territories. The formula was more likely to produce territory representation with population closer to the national quota than can be produced by the current method defined by Section 24.

As shown by the above example for South Australia in 1967, Dean’s method can be argued as providing a more proportional method for allocating seats to states than Webster’s Section 24 method.

If that is the case, can this overcome the fact that on occasions Dean’s method might allocate an extra seat compared to Webster’s method in Section 24.

I’ll leave it to learned constitutional lawyers to argue that point, but repeat Justice Stephen’s observation that proportionality was an absolute condition, “as near as practicable” was in qualified terms.

As the relevant formula in Section 24 applies until “the parliament otherwise provides”, Dean’s formula can be adopted. The rounding-up method applied by the Representation Act was ruled unconstitutional in McKellar’s case as it was clearly not more proportional. Using Dean’s method would have a strong chance of withstanding any High Court challenge.

The 1961 Apportionment

The result of the 1961 apportionment was controversial as it would have stripped both Western Australia and Queensland of a seat. When the Country party threatened to combine with Labor to defeat the redistribution, the Menzies government withdrew the legislation. At the time legislation for electoral boundaries needed to pass parliament before a new apportionment could be implemented.

Table 3 below sets out the calculations based on the population figures published by the Chief Electoral Officer. Tasmania has been left out of the table as it was guaranteed five seats. The quota in 1961 was 86,797.

Table 3 –  The 1961 Apportionment

Population Quotient Webster Dean
NSW 3,916,907 45.1272 45 45
VIC 2,930,244 33.7597 34 34
QLD 1,513,625 17.4387 17 17
SA 969,339 11.1679 11 11
WA 735,199 8.4703 8 8

Dean’s method would have made no difference in this case. Queensland would have lost a seat by both methods, and Western Australia would have required a quotient of 8.4706 for a ninth seat under Dean’s method. Western Australia was 26 voters short of a ninth seat under Dean’s method, and 2,578 short of a ninth seat by Webster’s method. In the end, under all possible methods, a minor difference in calculation can produce a change in representation.

Also note that until 1967, Australia’s Constitution included the following provision as Section 127.

In reckoning the numbers of the people of the Commonwealth, or of a State or other part of the Commonwealth, aboriginal natives shall not be counted.

Western Australia has always had a large indigenous population. The state would not have lost its ninth seat in 1961 were it not for this colonial, some would say racist, provision of the Constitution. Section 127 was removed by referendum in 1967.

In Conclusion

For the inquiry into representation of the Northern Territory in the House of Representatives, I will make the following recommendations.

  • The parliament should adopt the Dean method’s harmonic mean test in place of the Section 24 method incorporated into Section 48 of the Commonwealth Electoral Act. Using Dean’s method produces a seat allocation that will produce a Territory population per member average closer to the National quota than the existing method.
  • If parliament does fix the NT’s (or territories’) minimum representation at two seats, it should also adopt the Dean method’s harmonic mean test for allocating additional seats above the minimum.
  • If the Dean method’s harmonic mean is adopted, the extra provisions concerning statistical error should be dropped from the territory allocation formula.
  • Parliament should also adopt the Dean method’s harmonic mean for allocating seats to states in place of Section 24’s arithmetic mean. It will only rarely alter the allocation of seats, in only 1 of 26 past apportionments  as set out above. It can be argued as being more proportional than the existing method, and can therefore survive the tests previously set out in McKellar’s case (1977).


Determination of Entitlement of Federal Territories and New States to Representation in the Commonwealth Parliament“, Parliament of the Commonwealth of Australia, Joint Select Committee on Electoral Reform, Report No.1, November 1985. AEC submissions to this Committee were the source of past population and apportionment decisions prior to 1984 used in this blog post.

Territory Representation – Report of the Inquiry into increasing the minimum representation of the Australian Capital Territory and the Northern Territory in the House of Representatives“, Parliament of the Commonwealth of Australia, Joint Standing Committee on Electoral Matters, November 2003. AEC subissions to this committee were the source of past population and apportionment decisions 1984-2003 used in this blog post.

Further Reading

If you are really interested in the problem of apportioning seats to states and territories, especially in the United States, the most thorough (but be warned mathematically tough) exposition can be found in –

Fair Representation: Meeting the Ideal of One Man, One Vote“, Michael L Balinski and H Peyton Young, 4th Edition, Brookings Institution Press, 2001

Leave a Reply