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Transposition of Votes – 2013 Representation Order

3. Methods

The objective of the transposition of votes process is to determine how the 2011 general election voting results are to be distributed among the new FEDs under the 2013 Representation Order. This section provides an overview of the methods used by Elections Canada to determine the registered parties' candidates who have the right to provide the returning officer with lists of qualified persons to be appointed as election officers for the first general election that will be held under the new Representation Order.

As the information available for transposition varies by type of poll (ordinary poll, advance poll or mobile poll) or voting method (Special Voting Rules), a slightly different approach was used for each type.

3.1. Votes transposed for ordinary polls

Simply put, the transposition of votes consists of deriving the voting results for each political party at the 41st general election in each FED under the 2013 Representation Order. These calculations are straightforward for FEDs with unchanged boundaries from the 2003 Representation Order to the 2013 Representation Order since they involve only a restatement of the 2011 general election voting results. For the other FEDs, the voting results must be derived for each part of the old FED that composes the new one, and then added up.

Calculations involve the following steps:

Geographic superimposition

The geographic superimposition aims at establishing links between the four maps: the old and new FEDs, the set of PDs used at the 41st general election, and a map version of the list of electors.Footnote 3 This step consists of a series of cartographic overlays to determine how the PDs of the old FEDs are to be reassigned to the new FEDs. This process identifies not only the PDs that are moved in their entirety to one FED or another, but also those that are split by the new boundaries. These overlays also allow linking between the new FEDs and the electors' physical addresses.Footnote 4 At the end of the overlay process, each elector's physical address on the list of electors is precisely geo-located within a PD, as well as within old and new FEDs as shown in Figure 1 below. Therefore, the number of electors on that list can be counted for each new FED-PD piece.

Calculations of the ratios

After the old and new FEDs, PDs and electors have been linked, a ratio R can be calculated for each PD. Thus, R represents the proportion of electors assigned from that PD to a given new FED, for example F. It is obtained by E/T, where E is the number of electors in the PD assigned to F and T is the total number of electors in the PD. R varies between zero and one; R=0 means there are no electors assigned to the new FED and R=1 means all electors are assigned to the new FED. The calculation of R is performed at the PD level because voting results for ordinary polls are available at this geographical level.

Figure 1 – Geo-localization of electors' physical addresses within PDs, old FEDs and new FEDs (example)


Text version of Figure 1

Votes transposed for each political party to each new FED

Let V be the number of ballots cast for a given political party in a given PD at the 41st general election; the number of votes transposed for that party to a given new FED is defined by RxV. Therefore, the number of votes transposed for a given political party to a given new FED corresponds to the sum of RxV for that party in all PDs (entirely or partially) assigned to the new FED.

The steps described above are applied to the ordinary PDsFootnote 5. However, they are not directly applicable for the other types of polls, such as mobile polls and Special Voting Rules,Footnote 6 as the voting results are not available at the polling division level. The approaches used to transpose the number of votes for these types of polls differ slightly and are discussed hereafter.

3.2. Votes transposed for advance polls

Each advance poll is composed of a set of ordinary polls. As discussed in section 3.1, some ordinary polls may be split by new FED boundaries, which in turn will make their corresponding advance poll be split by the same boundaries. In other instances, a new FED boundary may follow PD boundaries within an advance poll, resulting in a split. In such cases, the ballots of the advance poll can be allocated to the new FED using the votes transposed for the corresponding ordinary polls.

Therefore, the number of votes transposed for a given political party of an advance poll corresponds to the addition of the votes transposed for that party in all the ordinary polls that compose the advance poll of the new FED.

3.3. Votes transposed for mobile polls

A mobile poll is one staffed by a deputy returning officer and a poll clerk, who travel on election day from institution to institution where seniors or persons with disabilities reside, to take their votes. When a mobile poll is split by the new boundaries, the number of institutions on each side of the boundary is used to calculate a ratio that is applied to the votes collected at the mobile poll.

3.4. Votes transposed for Special Voting Rules

Special Voting Rules (SVR) votes are ballots cast by Canadian Forces, incarcerated and international electors, and national and local electors voting by mail or at a returning office. The number of SVR votes in the 41st general election was 279,355. Unlike the transposition of votes completed in 2003, the 2013 transposition of votes includes SVR votesFootnote 7. Including SVR votes will ensure that all votes are taken into account for the transposition and that the total number of votes transposed corresponds to the number of official ballots cast.

Unlike votes cast by ordinary ballots, SVR votes for the 41st general election are available only at the old FED level. Therefore, no information is available at the PD level. To transpose the SVR votes of the 41st general election to the new FEDs, the ratio calculated for each part of the old FEDs by using the transposed electors on lists (as defined in section 3.5 below) is applied to the SVR votes cast for each political party.

3.5. Electors on lists transposed

As for all federal elections, the official voting results of the 41st general election were published along with the number of electors on lists. The number of electors on lists transposed to the new FEDs is also published with the transposed votes. Transposed electors on lists are derived in a similar manner as the votes.

Assuming that E is the number of electors on lists in a given PD at the 41st general election, the number of electors on lists transposed to a new FED is obtained by RxE; R is defined in section 3.1. The number of electors on lists in a new FED is obtained by adding RxE for all PDs (entirely or partially) transferred to the new FED.

For Canadian Forces, incarcerated and international electors who cast ballots under the SVR, the same ratio used to transpose the SVR votes (as defined in section 3.4) is also used to derive the number of electors on lists for the new FEDs. National and local electors who cast ballots under the SVR are excluded from these calculations as they are already included in ordinary poll elector counts.

3.6. Ranking of registered political parties

To determine the registered parties' candidates who have the right to provide the returning officer with lists of qualified persons to be appointed as election officers for the first general election that will be held under the 2013 Representation Order, the registered political parties (at the 41st general election) should first be ranked in each new FED. This ranking exercise is done after the votes for all types of PDs have been transposed. Candidates of political parties ranked first or second will have that right. These parties are those that counted the two highest numbers of votes transposed. In case of a tie, both parties are considered to be ranked in first and second places. In that event, they would appear in the relevant tables as Rank 1: Party A/Party B; and Rank 2: Party A/Party B.

3.7. Population

The total population counts presented in this report are based on the 2011 Census of Population by Statistics Canada.

3.8. How votes are treated

The reassignment allows for the transposition of all votes from the old to the new FEDs. The number of votes transposed for each officially registered party at the 41st general election is calculated using the following rules:

3.9. Remarks

Different approaches were considered for transposing the results of the 41st general election.

The approach used in 2003 for the transposition of the 2000 general election voting resultsFootnote 8 was based on voting results at the polling division level. To allocate votes when polling divisions were split by new FED boundaries, the voting age population and the geographic area information were used to calculate ratios to allocate votes accordingly.

The approach selected for 2013 was also based on polling division voting results. The votes transposed are direct restatements of the voting results for the vast majority of polling divisions (69,335 PDs out of 70,852 or 97.9 percent). To allocate votes to new FEDs when polling divisions were split by the new FED boundaries (2.1 percent of PDs), electors' addresses from the list of electors were used. This procedure includes a geo-localization of electors' physical addresses within polling divisions. This approach assumes that the voter participation rate is the same for each portion of split polling divisions. As the rate may be different in certain areas, the transposed results could be less accurate for these areas.

Transposed votes were validated throughout all steps of the transposition processFootnote 9. Control measures were put in place to minimize errors, such as misclassification of addresses. Misclassification errors occur when an elector's physical address is located on the wrong side of a new FED boundary. This can occur for a number of reasons, such as the quality of the elector's address information and misalignment between some geographic layers.


Footnote 2 In this document, the set of federal electoral districts in effect at the 41st general election are referred to as old FEDs.

Footnote 3 A map version of the list of electors is a point layer; each point represents an elector’s physical address location with the number of electors for that address attached to it. The list was extracted from the National Register of Electors in September 2013.

Footnote 4 As opposed to a mailing address.

Footnote 5 The 41st general election included 70,852 PDs, comprised of 64,501 (91 percent) ordinary polls, 1,645 advance polls (2.3%) and 4,706 mobile polls (6.7%).

Footnote 6 In this document, Special Voting Rules are considered as a PD type.

Footnote 7 For the 2003 transposition of votes, SVR votes were not transposed but were reported at the provincial level.

Footnote 8 Chief Electoral Officer of Canada. Transposition of Votes: 2003 Representation Order. December 2003. EC 60338 (12/03).

Footnote 9 Elections Canada would like to thank Sander Post from the Social Survey Methods Division at Statistics Canada for providing comments on the methodology.