6. The Sixth “Population Estimates” Formula (2011-2022)


Spreadsheets

  • A
  • B
  • C

This spreadsheet is entitled “1. MPs under the 6th Formula, 2010s” and breaks down as follows.

Descriptions

Sheet 1 starts off with the number of MPs per province under the Sixth Formula in section 51(1) of the Constitution Act, 1867 and the Population Estimates of 1 July 2011, which ultimately culminated in the number of MPs per province listed under the Representation Order, 2013. I abbreviated this sheet “RO, 2013 (Operative).” Only sheet 1 contains the real, operative figures under the Representation Order, 2013 and the real calculation done by the Chief Electoral Officer for the 2010s.

Sheet 3 (“2010s, Using Censuses”) shows a hypothetical Sixth Formula which used the decennial census instead of the population estimates for 1 July in the year of a decennial census to calculate the number of MPs per province, all while maintaining the initial federal electoral quotient of 111,166. Sheet 4 (“2020s, Using Censuses”) extrapolates that hypothetical Sixth Formula using the decennial censuses of 2011 and 2021 to calculate the average rate of growth a a new (lower) federal electoral quotient for the 2020s and the number of MPs per province in the 2020s.

Sheet 5 (“2010s Estimates for Q2 2001”) explores another possibility mentioned in the endnotes in chapter 7 of the book, where Elections Canada really should have used the estimated populations of the provinces for 1 July 2001, instead of the censal populations of the provinces for 2001, in calculating the first instances of the Representation Rule in 2011. Thankfully, however, the two sets of figures (the estimated populations for 1 July 2001 versus the censal populations for 2001) produced the same answers relative to the estimated populations for 1 July 2011. I therefore kept using the censal populations for 2001 in my main Sheet 1 (RO, 2013(Operative)), even if the text of the Population Estimates Formula suggests otherwise.

Sheet 6 (“2010s Quotient) carries over this thoughtexperiment and reverse engineers the initial federal electoral quotient of 111,166 in the 2010s, which the preamble of the legislation enacting the Sixth Representation Formula described as the average population per riding in 2001 based on the 295 federal electoral districts of the ten provinces under the Representation Order, 1996 and the estimated populations of the provinces for 1 July 2001 multiplied by their average growth relative to the estimated population of the provinces on 1 July 2011 and the 305 federal electoral districts for the ten provinces under the Representation Order, 2003. The artificial federal electoral quotient of 103,397 in the 2000s multiplied by the average rate of growth between the estimated populations of the provinces on 1 July 2001 and 1 July 2011 produced an artificial federal electoral quotient of 111,545 for the 2010s – very close but not exactly same of the real figure of 111,166 that Parliament assigned for the 2010s.

The Names and Estimated Populations

Column A names the provinces and territories. Into column B, I copy-pasted the population estimates for 1 July 2011.

Rule 1

Rule 1 of section 51(1) instructs us to divide the estimated population of the province by the federal electoral quotient for the 2010s of 111,166 (which comes from rule 6(a) of the Constitution Act, 1867), and then round up the raw number to the next larger integer. Column C therefore applies the federal electoral quotient to all ten provinces. Column D contains the formula (=b4/c4, etc.) and the results of that division. Column E then applies the formula (=roundup(D4,0)), which means rounding the raw number in column D to the next larger integer with 0 decimal places. I highlighted these headings in light green.

Rule 2

Rule 2 contains two instructions: first, apply the Senatorial Clause of section 51A such that no province ends up with fewer members in the House of Commons than it has in the Senate; second, apply the Grandfather Clause such that no province ends up with fewer members in the House of Commons in the 2010s than it held when the Constitution Act, 1985 (Representation) entered into force in 1986. Strictly speaking, based purely on the arithmetic, we could skip the Senatorial Clause because the Grandfather Clause captures the same information, but these two provisos fall under different constitutional amendment formulas, and the precise breakdowns remain instructive and interesting.

Senatorial Clause

I applied the Senatorial Clause in Columns F, G, and H and highlighted these headings in light red. First, I entered manually the number of senators per province under section 22 of the Constitution Act, 1867 in Column F. Second, Column G shows the difference between the number of senators in column F and the number of MPs in column E through the following formula: =IF((F4>E4),(F4-E4),0)

Only for the four Atlantic Provinces does that difference equal an integer of 1 or greater; for the other seven provinces, the formula produces 0. Column H then simply adds the number of MPs under column E and the number of MPs added under the Senatorial Clause to show the sub-total under this half of Rule 2: =E4+G4, etc.

Grandfather Clause

I applied the Grandfather Clause in columns I, J, and K and highlighted these headings light blue. First, I entered manually in column I the number of MPs per province when the Constitution Act, 1985 (Representation) entered into force 1986, which reflects the numbers under the Representation Order, 1976. Second, column J calculates the difference between the number of MPs under columns H and E under the same sort of if-then formula as under the Senatorial Clause: =IF((I4>H4),(I4-H4),0). The results of the logical test awarded additional MPs to Newfoundland & Labrador and to New Brunswick even on top of the Senatorial Clause, as well as additional MPs to Quebec, Manitoba, and Saskatchewan for the first time under Rule 2. Column K then displays the sub-total of MPs under the Senatorial Clause in column H plus the number of MPs added under the Grandfather Clause in column J: =H4+J4

This sub-total therefore completes both parts of Rule 2 and will feed into the subsequent calculations under Rules 3 and 4.

Rules 3 and 4: The Representation Rule

Rules 3 and 4 set up a convoluted logical test that can only possibly apply to Quebec: if a province was over-represented in the House of Commons compared to its share of the total provincial population under the previous electoral readjustment, then it cannot become under-represented under the application of Rules 1 and 2 in the current readjustment. And if said province does become under-represented under rules 1 and 2 in the current readjustment, then it must obtain the additional number of MPs necessary to bring its share of the House of Commons “as close as possible to, without being below” its share of the total provincial population, which would make this province either perfectly represented or slightly over-represented in the final calculation. A province must meet both of these conditions to obtain additional MPs under this rule. So while Ontario, Alberta, and British Columbia remain under-represented in the 2010s, they do not get more MPs because they were also under-representation, not over-represented, in the 2000s.

I made the headings for Rules 3 and 4 light yellow.

First Part of the Logical Test: Was the Province Over-Represented under the Previous Electoral Readjustment?

Rules 3 and 4 require 12 separate columns to illustrate each necessary step in the calculation per column. First, columns L, M, N, O, and P calculate whether each province was over- or under-represented in the 2000s to obtain the answer to the first part of the logical test.

In column L, I manually entered the number of MPs per province under the previous representation order, in this case that of 2003. In column M, I then manually entered the population under the decennial census of 2001, since the 279 Formula used that standard. Column N then calculates each province’s share of the population in the 2000s by dividing its population by the total provincial population in the 2000s: =M4/M$14.

The cashsign acts as an anchor in Excel and Calc when you extend a formula over into subsequent rows and columns; in this case, I want to proceed to M5 but keep the same reference in M14 instead of also moving down to M15.

Column O likewise calculates the share of MPs held by each province under the Representation Order, 2003 as: =L4/L$14

Column P then shows the difference between columns O and N, in that order: =O4-N4

The Second Part of the Logical Test: Is the Province Now Also Over-Representation Under the Current Electoral Readjustment?

This half takes up columns Q, R, S, T, U.

Columns Q, R, and S repeat the same steps as under columns N, O, and P under the first half of the logical test, but measure each province’s share of the population under the population estimates for 1 July 2011 and its share of MPs under Rules 1 and 2, and then computes the difference. The differences in column S show that Quebec, Ontario, Alberta, and British Columbia would be under-represented in accordance with Rules 1 and 2.

The results in column S show that Quebec would join the ranks of Ontario, Alberta, and British Columbia in being under-represented in the 2010s.

Column T then multiplies the percent difference in column S by what Rules 3 and 4 call “the multiplier,” which is to say, the sub-total number of MPs under the application of Rules 1 and 2, shown in cell K14 in this sheet, which is to say, 332: =S4*K$14

The product in column T then produces a positive number of the how many extra MPs an over-represented province has presuming representation by population in a House of Commons where 332 MPs represent the ten provinces; alternatively, it produces a negative number of how many more MPs an under-represented province would need, presuming a House of Commons capped at 332 MPs. The instructions under Rule 3 that the number of new MPs assigned be “as close as possible to, without being below” also require rounding up that raw number, as expressed in column U: =ROUNDUP(T4,0)

Combining the Two Logical Tests Together

Column V then combines these two logical tests together with the following formula. A province needed to obtain differences between its share of the House of Commons greater than 0 in the 2000s (column P) but less than 0 in the 2010s (column S) to qualify for the additional MPs in column U – in other words, to be over-represented in the 2000s but under-represented in the 2010s. Only Quebec meets both logical tests, and it therefore obtains an additional three MPs in column V, which applies the absolute value of column U if both conditions are met:

=IF(AND((P4>0), (T 4>0)), U4, 0)

=IF(AND((P4>0), (S4<0)), ABS(U4), 0)

Finally, column W adds up the sub-totals under Rule 2 and Rules 3 and 4 together to obtain the total number of MPs per province in the 2010s: =K4+V4

The Separate Territorial Representation Clause

Section 51(2) of the Constitution Act, 1867 assigns each of the three territories one MP irrespective of their populations, given that their combined populations approximately equal the federal electoral quota applied to the ten provinces and would grant the three territories combined only one MP if they fell under section 51(1).

The Grand Total

Column Y lists the grand total number of MPs in the House of Commons, replicating the provincial sub-total in column W and the territorial sub-total in column X.

The Sixth “Population Estimates” Formula, 2020s: Sheet 2 (MPs in 2021 (Abandoned))

Rule 6(b)

Columns B, C, and D contain Rule 6(b) shows how to calculate the new federal electoral quotient for the 2020s, which equals the average growthrate between the Population Estimates for 1 July 2011 and those for 1 July 2021 multiplied by the previous federal electoral quotient for the 2010s. I copy-pasted those estimated populations for 1 July 2011 and 1 July 2021 manually in columns B and C and calculated in column D the growthrate per province by dividing the new estimated population by the old estimated population:

=C4/B4

The average then equals:

=AVERAGE(D4:D13)

The new federal electoral quotient then comes to the product of the average growthrate and the old federal electoral quotient:

=B16*D15

The federal electoral quotient increased from 111,166 in the 2010s to 121,891 in the 2020s. These calculations in Sheet 2 (“Electoral Quotient, 2020s”) remained intact even after Parliament amended the Grandfather Clause in June 2022.

Rule 1

Columns E, F, and G contain the calculations under Rule 1. First, column E copies the new federal electoral quotient for the 2020s (121,891) for each province. Second, column F then displays the raw number of MPs of each province derived from the estimated populations divided by the federal electoral quotient: = C4/E4.

Third, column G shows those results rounded up to the next larger integer and equals the baseline number of MPs per province before all the distortions to representation by population enter into the equation: = ROUNDUP(F4, 0)

Rules 2, 3, and 4

All the other calculations under Rules 2, 3, and 4 (the Senatorial and Grandfather Clauses and Representation Rule) remain the same as in Sheet 1.

But Quebec did not grow quickly enough to keep its three extra MPs under the Representation Rule and instead went down to two.

Counterfactuals on the Sixth “Population Estimates” Formula in the 2020s

These worksheets form part of “1. MPs Under the 6th and 7th Formulas, 2010s-2020s.”

Sheet 4: MPs in 2021 (Bloc Bill)

Rule 4.1 under Bill C-246

I performed what seemed like the most logical calculation under what Bill C-246 would have established as Rule 4.1 by adding in columns N, O, P, Q, R, S, and T – which shifted the calculations under Rules 3 and 4 to the right relative to Sheet 3 “MPs in 2021 (Abandoned).”

MPs never even got the change to debate the sheer arithmetic absurdities which abound in Champoux’s constitutionally invalid bill. First, the numeration of this rule as “4.1” made no sense, because the rest of the paragraph places the calculation under rule “4.1” immediately after rule 2 rather than after rule 4, which means that it should appear as “2.1” instead. Second, this proposed  rule 4.1 bill would also have introduced at least two strange arithmetic inconsistencies into the Representation Formula. One, the rest of rules under section 51(1) use the total number of provincial MPs, not the grand total of all provincial and territorial MPs, as their baseline for calculating the number of MPs per province because Parliament already provided separately under section 51(2) that each territory shall have one MP irrespective of their populations. In other words, Parliament exempted the territories from representation by population altogether and could do so because section 52 only mandates “the proportionate representation of the provinces” in the House of Commons. This sleight of hand of mashing together two different standards of sections 51(1) and 51(2) would in February 2022 have increased the denominator from the 337 provincial MPs to the grand total of all 340 MPs in the House of Commons. Two, the order of operations and the order in which the calculations are carried out under rules 1, 2, 3, 4, and 4.1 matter and can change the outcome – just as happened under the Third Formula in 1971.

Applying this rule 4.1 after rules 1 and 2 but before rules 3 and 4 means that Quebec would hold 75 out of 340 MPs, or 22.06%. One-quarter minus 22.06% comes to 2.94%, which, multiplied by those 340 MPs would give Quebec 10 more MPs. Logically, the first baseline of 340 would remain the divisor in calculating each province’s share of MPs, because 85 out of 340 comes precisely to 25%. However, the numeration 4.1 and the phrase “after the completion of the readjustment” probably means that calculation under rules 3 and 4 would still apply, which would still award Quebec two more MPs because rules 3 and 4 use different baselines than rule 4.1. Quebec would then hold 87 MPs in a House of Commons of 352 MPs, which only comes to 24.72% instead of 25%. Logically, rule 4.1 should use the baseline of 350 MPs derived before the separate calculations under rules 3 and 4, yet even then, 87 out of 350 MPs comes to only 24.86%. Since rule 4.1 insists that Quebec must not hold “less than 25% of the total number of members in the House of Commons,” this would seem to mean that the denominator must keep going up in each calculation, which creates a loop of escalating numerators and denominators which only the rules of normal rounding could close, such that 24.72% becomes 25.00%. Champoux’s bill should therefore also have repealed rules 3 and 4 to avoid this mess – which, ironically, Parliament alone could do under the Section 44 Amending Procedure.

Column N adds the Territorial Representation Clause under section 51(2), which I entered manually. Column O then replicates the sub-total under Rule 2 (in column M) and adds in the three territorial MPs for a provincial and territorial grand total of 340 MPs for the purposes of the rest of Rule 4.1. Column P contains the crucial point of departure from Rules 3 and 4 under the real calculation from October 2021. Here, “Share of MPs Before” counts the three territorial MPs and increases the denominator from 337 to 340:

=O4/O$20

Column Q then calculates the difference between 25.00% and the provinces’ shares in column N:

=0.25-Q4

In Column R, I applied a logical test that if a province’s population equals Quebec’s, then the extra MPs to attain 25% applies and equals the number in column O multiplied by the total in column M; and if not, then 0:

= IF(C4=8604495, Q4*O$20, 0)

Quebec gained 10 MPs under this calculation. Colum S then lists the sub-total of MPs under columns O and R:

=O4+R4

Column T then uses that number sub-total in column S and the grand total in column O to calculate each province’s new share of MPs, and this is where the Bloc’s formula really runs into problems because it should have, but did not, purport to repeal Rules 3 and 4:

= S4/O$20

Since the Bloc’s ill-conceived and unconstitutional bill instructed the calculation under Rule 4.1 to happen after those under Rule 2, but also did not purport to repeal Rules 3 and 4 despite naming the new rule Rule 4.1, the calculations under Rules 3 and 4 must still take place after those under Rule 4.1. This still gives Quebec two more MPs, for a total of 87 out of 352 instead of 85 out of 350. Column AE calculates a new sub-total of MPs per province under Rules 4.1 and Rules 3 and 4 (=S4+AD4), but then column AF checks the final share of MPs that each province holds using the new denominator of 352 because of the poor wording of the Bloc’s bill, which, in turn, reduces Quebec’s share from precisely 25.00% to 24.72%. To avoid a continuous loop of escalating numerators and denominators, I also took the liberty of applying a round up function to column AF to cut off the head of the snake and put an end to all the calculations.

= AE4/AE$21

Sheet 5: The Vis Amendment

Conservative MP Brad Vis proposed an amendment at PROC which instead of changing the point of reference of the Grandfather Clause from to the MPs per province held in 1986 to that in 2020 would have gotten rid of the first half of the logical test under Rules 3 and 4. This way, any province under-represented under Rules 1 and 2 would gain as many MPs as necessary to bring its share of the MPs for the provinces in line of its share of the total provincial population, irrespective of whether it had also been over-represented under the calculation the decade before.

I have shown this calculation in sheet 5 (“2021, Vis Amendment”). The truncated Rules 3 and 4 now only takes up columns N, O, P, Q, R, S, and T. The logical test in column Q for “MPs Added” now only includes one condition instead of two:

= IF((Q4<0), ABS(R4), 0)

The Vis Amendment would have allowed Ontario, Alberta, and British Columbia to access the Representation Rule along with Quebec. Ontario would have gained 10 more MPs; Alberta, 3; and British Columbia, 4. Quebec would still have gained 2 under this standard. This, in turn, would have increased the size of the House of Commons to 356 provincial MPs and 359 MPs in total.

Sheet 6: Amending the Representation Rule and Repealing the Grandfather Clause

This worksheet takes the calculations in Sheet 5 “The Vis Amendment”, which amended the Representation Rule, and combines then with repealing the Grandfather Clause. Rule 2 now only consists of the Senatorial Clause, and Rules 3 and 4 apply the same one-part logical test as under the Vis Amendment so that all under-represented provinces qualify for the Representation Rule, which under this scenario only compensates them for the distortions caused by the Senatorial Clause.