Sainte-Laguë allocation formula

The Sainte-Laguë formula is used to allocate seats to parties contesting the party vote that cross the threshold for allocation of winning at least one electorate seat or 5% of all party votes cast nationally.

The following describes the steps and then uses the 2002 general election result to illustrate the process.  If you want to see what would happen with different levels of party vote you can use the MMP seat allocation calculator [1].

The chief electoral officer takes the following steps:

1. Totals all party votes across the country and creates a table showing for each party their overall number of party votes, the percentage of party votes and the number of electorate seats won. 
2. Determines which parties are to be excluded because they have not crossed the threshold. Details of step two using 2002 data.
3. Applies the Sainte-Laguë formula to the party votes for all parties that were left after step 2.
4. Determines the total number of seats each qualifying party was entitled to by counting the number of quotients that party had in the highest 120.
5. Compares the total number of electorates won by each party with the number allocated to that party in step 4. Each party is then allocated enough party list seats to bring its total number of seats up to the number to which it was entitled.
6. Examines the party list for each party and deletes the names of any candidates who have won an electorate.
7. Allocates the list seats from step 5 to the highest placed candidates left on the appropriate party list.
8. Declares those candidates elected to Parliament by publishing a notice in the New Zealand Gazette and advising the Clerk of the House of Representatives of their names.

The process is prescribed by the Electoral Act 1993 s.191-s.193A [2].

Step 1 - Summary table

The Chief Electoral Officer drew up a table with the name of each party shown on the party vote, the number of party votes it won, the percentage of all party votes it won and the number of electorate seats won by it and its component parties. (Some minor parties not crossing the threshold are combined under the heading 'Other parties' for this demonstration.)

 

Step 2 - Delete non-threshold parties

The Chief Electoral Officer then excluded parties that were not eligible for a proportional share of seats by deleting any party that had not crossed the threshold by winning at least 5% of the total number of party votes and had not won at least one electorate seat under its own name or that of a component party. The Christian Heritage Party gained 1.35% of all the party votes and did not win any electorate seats, so all the party votes for the Christian Heritage Party and other parties in the same situation were excluded.  Although Jim Anderton's Progressive Coalition won less than 5% of all the party votes, it won an electorate seat and so qualified for a proportional share of all 120 seats based on its party votes.

 

Step 3 - Divide by sequential odd numbers

The Chief Electoral Officer then took the total party votes for each qualifying party and divided the figures by a sequence of odd numbers starting with 1 (1, 3, 5, 7, 9, 11, 13, 15 and so on shown in the far left column of the table).  The resulting numbers are called quotients.  The division continues until enough quotients had been found to allocate all 120 seats. The numbers to the right of the highest 120 quotients indicate their order from highest to lowest. The 121st quotient is shown in brackets.

View Table [3]

Step 4 - Find proportional seat entitlement

 

 

 

	Labour	National	NZ First	ACT New Zealand	Green Party	United Future NZ	Jim Anderton's Progressive Coalition
No. of quotients in highest 120	52	27	13	9	9	8	2

Step 5 - Fill from electorates then list

 

	Labour	National	NZ First	ACT New Zealand	Green Party	United Future NZ	Jim Anderton's Progressive Coalition
No. of electorate seats won	45	21	1	0	0	1	1
No. of list seats allocated(bolded in table under Step 3)	7	6	12	9	9	7	1

 

 

Registered Parties that gained at least 5% of all Party Votes or won at least one electorate seat	Party Votes	% of all Party Votes cast for parties eligible for a proportional share of seats	Number of electorate seats won
Labour Party	838,219	43.38	45
National Party	425,310	22.01	21
NZ First	210,912	10.92	1
ACT New Zealand	145,078	7.51	0
Green Party	142,250	7.36	0
United Future NZ	135,918	7.03	1
Jim Anderton's Progressive Coalition	34,542	1.79	1
Total	1,932,229	100.00	69

 

 

Registered Parties	Party Votes	% of all Party Votes	No. of electorate seats won
Labour Party	838,219	41.26	45
National Party	425,310	20.93	21
NZ First	210,912	10.38	1
ACT New Zealand	145,078	7.14	0
Green Party	142,250	7.00	0
United Future NZ	135,918	6.69	1
Jim Anderton's Progressive Coalition	34,542	1.70	1
Christian Heritage Party	27,492	1.35	0
Outdoor Recreation NZ	25,985	1.28	0
Alliance	25,888	1.27	0
Other parties	20,023	0.99	0
Total	2,031,617	100.00	69