(**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 representatives (seats) each state and territory will have at the next federal election.

As the numbers stand, it is expected that Victoria will gain a seat to 39 seats, and Western Australia will lose the 16th seat it gained in 2016. Most controversially, the Northern Territory will lose the second seat it has had since 2001.

This is the second of three posts on Australian apportionment. 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.

In this post I concentrate on the constitutional basis and history of territory representation, and in what ways Territories are treated differently from the states in allocating seats.

The Labor Party is proposing a bill to save the NT’s second seat by legislating that the Northern Territory have a minimum of two seats. The NT’s Country Liberal Party has expressed some support for the idea. As was the case with a similar bill when the NT’s second seat was marked for abolition in 2003, the bill will be the catalyst for a more detailed discussion of the issue.

In my opinion, it would be better to change the formula as it applies to the territories rather than return to fixing the number of seats. In technical terms, my proposal is that the allocation of extra seats should be determined by rounding at the harmonic mean of two alternate allocations rather than the current arithmetic mean. In the case of the Northern Territory, that would involve allocating a second seat if the quota calculation is above 1.33 rather than the current 1.50. This would almost certainly save the NT’s second seat for the next election.

If you don’t have time to read through all the detail in this post, click here to go to the tables showing how the proposed change formula would have applied to the NT and ACT at apportionments since 1991.

### The Basis for Territory Representation

The Commonwealth of Australia was formed by the federation of the six self-governing Australian colonies in 1901. At the time the Northern Territory was part of South Australia, and what was to be the Australian Capital Territory part of NSW.

The Northern Territory was handed to the Commonwealth in 1911, and the ACT excised from NSW the same year. Canberra had been designated as the future capital in 1908, though it did not replace Melbourne as the capital until the provisional Parliament House opened in 1927. The migration of government departments up the Hume Highway to make Canberra a real capital took much longer.

Section 24 of the Constitution sets out a formula to allocate representation to each of the states. (See discussion in my first post on the 2020 apportionment.) Whether there would be territory representation, and on what terms it should be allowed, was a power given to the Commonwealth Parliament by Section 122 of the Constitution. Parliament can grant representation to territories “on the terms it thinks fit”.

One of the oddities of Section 122 is that it does not include the constitutional guarantees concerning members and senators from the states. There is no guarantee that a territory Senator has only one vote. There is no limit to the number of territory members that the parliament may “see fit” to allow. Territory representatives do not have to be directly elected and there is no “nexus” between the number of territory House and Senate representatives. There have been arguments these limits should be dealt with by constitutional amendment, but there has been little appetite for pursuing them down the road of constitutional referendum.

The ACT and NT were granted representation in the House by separate acts of parliament, the NT in 1922, the ACT in 1948. Both members had only limited voting rights until the ACT was granted full powers in 1966 and the NT in 1968. The ACT was granted a second seat by legislation in 1973. The territories were allocated two Senators each in 1975. The Hawke government’s 1983 electoral reforms incorporated territory representation into the Commonwealth Electoral Act, further transfers resulting in the separate territory acts being repealed in 1990.

Following the 1984 election, the Joint Select Committee on Electoral Reform inquired into representation of territories and new states. It recommended a number of changes that were incorporated into the Electoral Act, the most important being that the allocation of representation to territories would be determined by the same formula that applied to states.

How that formula works is set out in my first post on apportionment, but I have repeated details of the 2017 determination below to aid further discussion on how the formula applies to the allocation of territory seats.

Other current provision on minimum representation for the territories that won’t be discussed in this blog post are –

- Both the NT and ACT are guaranteed one seat in the House.
- No other territory can have representation unless its population is greater than half a quota.
- If not granted separate representation, Norfolk Island and Jervic Bay are treated as part of the ACT, and Christmas and Cocos (Keeling) Islands treated as part of the NT.
- The ACT and NT are entitled to two Senators, a figure that can be increased to equal half its House representation once either territory has more than six House seats.

### How the Current Formula Works – the 2017 Determination

To explain the current formula, the table below sets out the calculations used to allocate representation to states and territories by the 2017 determination, published 31 August 2017. (Full details of the 2017 determination can be found on the AEC website.)

**The determination was based on the population of the Commonwealth (that is the original states) being 23,729,561. Dividing by 144 (twice the state Senators) gave a quota of 164,788.61806.**

Dividing the population of each state and territory by the quota gives a **quotient**.

The quotient is then used to allocate seats as follows –

- The number of seats equal to the integer part of the quotient (that is the number to the left of the decimal point) is allocated to each state and territory.
- All states where the fractional part of quotient is greater than 0.5 are allocated an extra seat.
- For original states, if the rounding produces fewer than five seats, then the minimum of five seats applies.

Another way to look at the allocation is to do step one and two in a single operation. Seats can be allocated based on “natural” rounding, rounded down to the **lower bound** if the fraction is less than 0.5, or rounded up to the **upper bound** if the fraction is greater than 0.5. **The allocation is a “natural” rounding of the arithmetic mean of the lower and upper bounds, that is (Lower Bound + Upper Bound) / 2.**

(That the formula uses an arithmetic means becomes important for discussing alternative formulas later in this post.)

Table 1 shows the 2017 calculation of quotients, seats allocated, as well as the change in seats allocated since the previous determination in 2014. There are explanatory notes below the table.

## Table 1 – The 2017 Representation Determination

Population | Quotient | Seats | Change | |
---|---|---|---|---|

NSW | 7,797,791 | 47.31996 | 47 | .. |

VIC | 6,244,227 | 37.89234 | 38 | +1 |

QLD | 4,883,739 | 29.63639 | 30 | .. |

WA | 2,567,788 | 15.58231 | 16 | .. |

SA | 1,716,966 | 10.41920 | 10 | -1 |

TAS | 519,050 | 3.14979 | 5 | .. |

ACT | 408,562 | 2.47931 | 3 | +1 |

NT | 247,512 | 1.50200 | 2 | .. |

- The quotients for Victoria, Queensland and Western Australia were rounded up to the upper bound, NSW and South Australia rounded down to the lower bound.
- Compared the previous determination in 2014,
**Victoria**gained a seat and**South Australia**lost a seat. **Tasmania**was only entitled to three seats by the calculation, but the Constitution guarantees five seats to it as an original state.- The population of the
**ACT**includes Jervis Bay and Norfolk Island. The population of the**Northern Territory**includes Christmas Island and Cocos (Keeling) Island. - In population terms, the
**ACT**fell short of a third seat with a quotient of 2.47931. As explained below, since 2003 a statistical error margin has been applied to territories in the determination. Twice the statistical standard error added 10,694 to the ACT total, lifting the ACT to a quotient of 2.54420 and entitlement to a third seat. - The
**NT**was entitled to a second seat on population alone. It would have been lifted to 1.547 quotas on standard error if it had fallen short of two quotas.

Estimates for the 2020 entitlement review were published by the Parliamentary Library in July 2019. These project WA to lose a seat and Victoria to gain another seat. The same paper projected the NT quotient to be 1.430 quotas. Even statistical error would not be enough to save the NT’s second seat. (Find the Parliamentary Library’s estimates at this link.)

### Adjustments for the Territories – Saving the NT’s Second Seat in 2004

Having been allocated a second seat ahead of the 2001 election, the Electoral Commissioner’s determination in 2003 was that the NT would revert to a single seat. The population was 199,760, 1.4978 quotas and short of the 1.5 figure that entitled a second seat. The Northern Territory fell short of a second seat by 294 people.

Then Country Liberal MHR for Solomon, Dave Tollner, proposed a bill that would restore the NT’s second seat and guarantee two seats in the future. The bill was referred to the Joint Standing Committee on Electoral Matters for an inquiry. The result was that rather than fix the number of members, the formula was adjusted to take account of statistical error in the population estimates.

For both the Northern Territory and ACT, the Australian Bureau of Statistics provides a value representing twice the standard error of the population estimate. If the quotient produced by the quota was below the cut-off value of 0.5 between the upper and lower bounds (1.5 in the case of the NT), but above 0.5 **after including twice the statistical error**, the territory could be allocated the seats for the upper bound (two in the case of the NT).

The error margin saved the NT’s second seat for the 2004 election, and allowed the ACT to be allocated a third seat in the 2017 determination. (How the standard error worked to allocate the ACT a third seat in 2017 is explained in the Table 1 footnotes.)

But estimates published by the Parliamentary Library in 2019 suggest the slower rate of NT population growth relative to other states could cut the NT’s quotient to only 1.43 quotas, well below where the second seat can be saved by the statistical error.

Under current legislation, the Electoral Commissioner’s determination of representation in early July is the final word. It will immediately set in train redistribution procedures for any state or territory with a change in representation. In the case of the Northern Territory, reversion to a single seat would be implemented by merging the two existing seats without a redistribution. Note that both current NT members would continue to represent their electorates in the current parliament, but there would be only on seat for them to contest at the next election.

As in 2003, if the current Parliament decides to undo the determination for the Northern Territory, it will require legislation to overturn the determination and restore the former electorates.

### Lessons from Apportionment In the United States

There is a vast literature on apportionment formula for allocating seats to states in the United State’s House of Representatives. The same methods under different names are also widely examined for allocate seats to parties for electoral systems based on proportional representation.

While useful as reference, the US literature has a number of assumptions that aren’t relevant in the Australian context. I’ll discuss some of these further down in this post.

The US apportionment experience is useful in the Australian context for discussing how to deal with representation for the territories. There is no constitutional bar on the Commonwealth Parliament using a different formula for allocating seats to territories, and the US debate offers many alternate formulas.

Discussing apportionment formula has to acknowledge there are significant scale issue between Australia and the USA in both number of states and number of members to aportion.

- The Australian House of Representatives has around 150 members to be allocated across six states and two territories. States are guaranteed a minimum five seats, territories one seat. Tasmania has the guaranteed five seats, the next smallest state South Australia currently has 10 seats. For the last three decades, the Northern Territory has switched between one and two seats, the ACT between two and three seats.
- The United States House of Representatives has 435 members to be allocated across 50 states. All states are guaranteed a minimum one seat. Many US states are small in population. Currently seven US states having a single member, five have two members, three have three members, and six have four members. Four states have more than 25 members and the largest state California has 53.

As with the Australia’s territories, the operation of the seat allocation formulas on small US states can be tough. When the number of seats is small, the average number of people per seat varies greatly depending on whether the number of seats allocated equals the quotient’s lower or upper bound.

Let me demonstrate this by presenting the 2017 determination data from the perspective of average people per seat. In Table 2, the final two columns show the average number of people per seat when a state is allocated the lower bound number of seats, and the average number when allocated the upper bound of seats. (Tasmania is left blank as it is guaranteed five seats.)

### Table 2 – Comparing Average Population at Upper and Lower Bounds

Seat Bounds | Average People | ||||
---|---|---|---|---|---|

Population | Lower | Upper | Lower | Upper | |

NSW | 7,797,791 | 47 | 48 | 165,910 | 162,453 |

VIC | 6,244,227 | 37 | 38 | 168,762 | 164,321 |

QLD | 4,883,739 | 29 | 30 | 168,404 | 162,791 |

WA | 2,567,788 | 15 | 16 | 171,185 | 160,486 |

SA | 1,716,966 | 10 | 11 | 171,696 | 156,087 |

TAS | 519,050 | .. | .. | .. | .. |

ACT | 408,562 | 2 | 3 | 204,281 | 136,187 |

NT | 247,512 | 1 | 2 | 247,512 | 123,756 |

For the largest state NSW, switching between the upper and lower bound number of seats creates a difference of 3,457 people per representative. In South Australia, a smaller state, the difference is 15,609. For the ACT it is 68,094, and for the Northern Territory 123,756.

The smaller the state or territory, the fewer the seats, the larger the average difference created by switching between the lower or upper bound of seat allocation.

Update:

An entirely different approach would be to allocate a number of seats to a state or territory **based on which of the upper or lower bound averages is closer to the national quota figure**. The 2017 national quota was 164,789.

If you compare the last two columns of Table 2 to the national quota, then in the five mainland states and the Northern Territory, the allocation of seats is exactly the same as by the Section 24 method.

There is a difference for the ACT though. Under the Section 24 method, the quotient was 2.47391 and would have allocated only two seats were it not for the additional test on applying an error margin. Using the closeness to national quota method, the ACT would be allocated three seats without resort to error margin. In table 2, the upper bound average of 136,187 was closer to the national quota 164,789 than the lower bound average of 204,281.

The method I’ve just described is known in the United States as Dean’s method. Turning this “closest to quota” criteria in mathematics, the test is to round up or down based on the fractional part of the harmonic mean between the upper and lower bound rather than the arithmetic mean used by Section 24. This minimizes the difference between a state or territory’s average population per seat and the national quota.

If that last paragraph causes your eyes to glaze over, let me use the numbers. For the NT’s situation, you should round up or down at a quotient of 1.33 rather than 1.50. For the ACT, you would round up or down on a quotient of 2.40 instead of 2.50. Rounding at this harmonic mean quotient value minimizes the difference between the national quota and the state or territory average.

Note also that the Section 24 method’s arithmetic mean is at 0.5 for every pairing of lower and upper bound. The harmonic mean varies depending on the value of the lower and upper bound, but converges on 0.5 as the integer value of the quotient increases.

At this point it is impossible to keep ignoring US literature on apportionment of seats to states. I am about to advocate using Dean’s method to allocate seats to states in Australia, but this is one of the apportionment methods that has never been used in the USA.

### Three American Methods – Webster, Huntington-Hill and Dean

Article 1 Section 2 Clause 3 of 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 to be 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. To introduce a few apportionment method names at the point, Webster’s method was used after the US Census of 1840, 1910 and 1930. Huntington-Hill (or Huntington, or Hill, or the method of equal proportions) has been used after every Census since 1940.

Before going into details of the methods, a few details of US apportionment process that do not apply to Australia are important to point out.

- As mentioned earlier, at 50 there are many more states in the USA, and 21 of the 50 states have fewer than five seats, the minimum number allocated to Australia’s six states.
- The US House of Representatives has been fixed at 435 seats for all apportionments in the last century. To make the outcome match the size of the House, the process has to be applied in iterations, multiple passes allocating seats rather than the one pass method applied by Section 24. The US formulas are now applied using divisors and allocation tables. (Explaining that last sentence gets a little beyond the purpose of this post so just trust me it is the same.)
- 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 means there is much more debate over whether a state has gained or lost a seat in relation to changes in another state’s allocation.
- 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 USA.

The Australian formula in Section 24 is the same as Webster’s method. For those interested in electoral systems, Webster’s method is the same as the formula used for the Sainte-Lague method of allocating seats in proportional representation. In Australia, Webster’s method allocates seats in a single calculation, in the USA it is applied in such a way that state seat numbers are juggled up or down to match the fixed House size.

The complexity of apportionments is that a state’s proportion of the national population is a real number (a number with decimals), but the number of seats it is 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 this case one or two.

The Webster, Huntington-Hill and Dean methods all start with the same formula to calculate the quota. The quota equals the population divided by the number of seats to be allocate. In the case of the US, the number is 435, in Australia twice the state Senators equals 144.

If applied in Australia, each of these methods would allocate the same number of whole quota seats at the first stage. All three methods allocate the integer (whole number) part of the quotient for each state or territory in Table.

Where the methods differ is in how they deal with the quotient fractions in Table 1. Each method uses a different rounding point to allocate seats to the upper or lower bound of seats.

To explain this, let me define two variables L and U.

**L is the lower bound**or naturally rounded down value of the quotient**U is the upper bound**or naturally rounded up value of the quotient

The difference between the methods in dealing with fractions are –

**Webster’s Method**– Extra seats are allocated where the remainder is greater than the arithmetic mean of the upper and lower bound. That is the test value (L + 0.5) or (L + U) / 2. This is the method in Section 24 of the Australian Constitution.**Huntington-Hill Method**– Extra seats are allocated for states where the remainder is greater than the geometric mean of the upper and lower bound, that is 1 sqrt(L*U)**Dean’s Method**– Extra seats are allocated based on the harmonic mean of the lower and upper bounds., that is 2*L*U / (L + U)

Table 3 below shows the rounding points for allocating the Lower or Upper bound number of seats. I have included values for states and territories between one and ten.

## Table 3 – Comparing the Methods

Lower Bound |
Upper Bound |
Webster | Huntington- Hill |
Dean |
---|---|---|---|---|

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 |

Looking at the Northern Territory which sits on a quotient between one and two seats, Webster’s method uses 1.5 as the threshold for a second seat, Huntington-Hill 1.4142, and Dean 1.3333.

For the ACT with a quotient between two and three, Webster’s method uses 2.5 as the threshold for a second seat, Huntington-Hill 2.4495, and Dean 2.4.

With the minimum five seats per state, the cut-offs for the three methods quickly converge to 0.5. The rounding points between five and six seats are 5.5, 5.4772 and 5.454.

South Australia currently has 10 seats, and with a population declining relative to other states, its next change in representation is likely to be down to nine seat. Under Webster’s method, SA would fall to nine seats if its quotient fell below 9.5, but the value is slightly lower for Huntington-Hill, 9.4868, and Dean 9.4737.

The differences between the three methods disappear the larger the state. For NSW on the divide between 47 and 48 seats, the rounding points are 47.5, 47.4974 and 47.4947.

The graph below plots the rounding points for the three methods between one and 20 seats and shows how all three methods converge towards 0.5.

The comparative literature in the United States puts great importance on measuring whether these formulas advantage large or small states. With so many states, so many small states, and iterative methods to fix the size of the House, Dean’s method has never been used in the United States for reasons I’ll move on to in my third post.

All I will say in this post about my choice of Dean’s method and the harmonic mean is that it is the method that sets to minimize the difference in average people per representatives. In the Australian context, I think that is the best difference to minimize to produce proportionality.

But the point of this post is to look at the application of these methods to the low population territories where the rounding points between the three methods is greatest. Let me turn to applying the three methods to the two territories.

### Applying Alternate Seat Allocation Methods to the Northern Territory

Under the existing seat allocation formula, the Northern Territory was first granted two seats for the 2001 election, was saved from reverting to one seat in 2004 by adding the standard error adjustment, but has been allocated two seats at every election since.

Applying the three methods, the Northern Territory is qualified for a second seat at 1.5000 under the current formula (Webster’s), at 1.4142 under the Huntington-Hill geometric mean method, and 1.3333 under Dean’s harmonic mean.

Table 4 below shows the number of seats allocated by each method since 1991, which was the first determination where territories were assessed using the Section 24 formula. Determinations where there is a difference between the methods are shown shaded, with the different number of seats in bold.

### Table 4 -Northern Territory Alternative Seat Allocations

Determination Election |
Quota | Arithmetic Seats |
Geometric Seats |
Harmonic Seats |
---|---|---|---|---|

1991/1993 | 1.3773 | 1 | 1 | 2 |

1994/1996 | 1.4284 | 1 | 2 |
2 |

1997/1998 | 1.4540 | 1 | 2 |
2 |

1999/2001 | 1.5239 | 2 | 2 | 2 |

2003/2004 | 1.4978 | 1 | 2 |
2 |

2006/2007 | 1.5054 | 2 | 2 | 2 |

2009/2010 | 1.5362 | 2 | 2 | 2 |

2011/2013 | 1.5263 | 2 | 2 | 2 |

2014/2016 | 1.5572 | 2 | 2 | 2 |

2017/2019 | 1.5020 | 2 | 2 | 2 |

**Compared to the current method, Huntington-Hill’s method using the geometric mean would have deliver the Northern Territory a second seat for the 1996 and 1998 elections, and the Dean method’s harmonic mean would have delivered a second seat for the 1993 election as well.**

**Both the geometric and harmonic mean tests would have retained the NT’s second seat at the 2003 determination without having the rely on the standard error adjustment.**

### Applying Alternate Seat Allocation Methods to the ACT

Under the existing seat allocation formula, the ACT has had three seats at only two elections, in 1996 and in 2019. The ACT passed the current 2.5 quotient to be granted a third seat for the 1996 election, but fell short for the 2019 election, only to be allocated the third seat based on the standard error adjustment.

Applying the three methods, the ACT is qualified for a third seat at 2.5000 under the current formula, at 2.4495 under Huntington-Hill’s geometric mean, and 2.4000 under Dean’s harmonic mean.

Table 5 below shows the number of seats allocated by each method since 1991, which was the first determination where territories were assessed using the Section 24 formula. Determinations where there is a difference between the methods are shown shaded, with the different number of seats in bold.

### Table 5 – ACT Alternative Seat Allocations

Determination Election |
Quotient | Arithmetic Mean |
Geometric Mean |
Harmonic Mean |
---|---|---|---|---|

1991/1993 | 2.4656 | 2 | 3 |
3 |

1994/1996 | 2.5042 | 3 | 3 | 3 |

1997/1998 | 2.4947 | 2 | 3 |
3 |

1999/2001 | 2.4254 | 2 | 2 | 3 |

2003/2004 | 2.4209 | 2 | 2 | 3 |

2006/2007 | 2.3751 | 2 | 2 | 2 |

2009/2010 | 2.3858 | 2 | 2 | 2 |

2011/2013 | 2.3849 | 2 | 2 | 2 |

2014/2016 | 2.4392 | 2 | 2 | 3 |

2017/2019 | 2.4793 | 2 | 3 |
3 |

**Compared to the arithmetic mean, the Huntington-Hill’s geometric mean would have granted the ACT a third seat at three extra elections, in 1993 and 1998, and in 2019 without having to rely on the error margin adjustment.**

**The comparison with the harmonic mean would have delivered three seats to the ACT at six extra elections, 1993, 1998, 2001, 2004, 2016 and 2019.**

The ACT under all methods would have had only two seats for the 2007, 2010 and 2013 elections.

### Recommendations

To save the Northern Territory’s second House of Representatives seat, and to deal more fairly with both territories, I would make the following recommendations on the formula used to allocate seats to territories.

- If parliament chooses not to fix the NT’s minimum House representation at two seats, it should instead adopt the Dean method’s harmonic mean test in place of Section 24’s arithmetic mean to provide a fairer method of proportionality.
- 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 as a fairer method of proportionality.
- If the Dean method’s harmonic mean is adopted, the extra provisions of statistical error should be dropped from the territory allocation formula.

### Sources

“*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 these blog posts.

“*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 these 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