Spreadsheets
- A
- B
- C
Description
The Second Formula could be called the Fixed Floor Formula because it locked the House of Commons at 254 provincial MPs in 1946 and later at 261 provincial MPs when Newfoundland joined Confederation in 1949. Rule 1 set the federal electoral quotient as the population of the provinces divided by 254, and divided each province’s population by that quotient to obtain the number of MPs per province, rounded down. If rule 1 produced fewer than 254 MPs, then rule 2 ensured that any extra MPs went to the provinces with the largest remainders in descending order until the total number of provincial MPs hit 254. But if applying rules 1 and 2 awarded any province fewer members of the House of Commons than it holds in the Senate, then rules 3 and 4 tied the Senatorial Clause into the calculation. After subtracting both the populations and MPs of provinces which ended up with fewer MPs than senators under the first round of rules 1 and 2, rules 3 and 4 would repeat the calculations done under rules 1 and 2 using these smaller populations and electoral divisor. In the 1940s, rules 3 and 4 applied only to Prince Edward Island, which reduced the total number of provincial MPs from 254 to 250, and any extra MPs left over at this stage went to the provinces with the largest remainders in descending order until the total number of provincial MPs hit 250. Order matters, and applying rules 3 and 4 means that rules 1 and 2 must be re-applied in a second round.
I have used sheet 1 (“RA, 1947”) as the example here.
Rule 1
I shaded the columns under Rule 1 (B, C, D, and E) as light green. In Column B, I entered the populations of each of the provinces by hand from the latest decennial census and the electoral divisor in column C (C13). Column D calculates the federal electoral quotient by dividing the total population of the provinces by the electoral divisor.
=B$13/C$13
Column D calculates the number of MPs per province:
=B4/D4
Column E then rounds down that result to the lower integer:
=ROUNDDOWN((D4),0)
The sum of MPs rounded down (=SUM(E4:E12)) came to only 250, which means that the remainders add up to 4, for the total of 254 MPs.
Rule 2
Rule 2 takes up columns F, G, and H in blue. Ideally, I would have liked to apply an if-then function to column G such that if column G is grey, then G equals 1; if not, then G equals 0. However, that would have required learning a bit too much about programming Excel or Calc for my liking, so I simply entered “1” manually on the provinces with the four largest remainders, highlighted in grey based on the conditional formatting described in the previous step.
Column F lists the remainders from column E using a TRUNC function.
=D4-TRUNC(D4)
I also applied a conditional formatting to highlight the four largest remainders in the column in grey. In Calc, I selected the Format drop-down menu, followed by Conditional, Condition, More Rules, and “cell value” “is equal to” the sum of the remainders (in this case, 4). I had to change this value manually in each decade as required.
Column H then finds the sub-total number of MPs per province under Rules 1 and 2 by adding up the number of MPs rounded down and the remainders:
=E4+G4
While the sub-total under Rules 1 and 2 adds up to 254, we cannot end our overall calculation here because of the Senatorial Clause. Prince Edward Island only ended up with two MPs under pure representation by population but needs four.
Rule 3
That dastardly Senatorial Clause, the bane of pure representation by population, rears its ugly head in light red in columns I, J, and K.
I entered by hand the number of senators per province under section 22 of the Constitution Act, 1867 in column I. Consequently, column J then lists the number of additional MPs that a province would need so that it ends up with at least the same number of MPs as senators with the following if-then function:
=IF((H4<I4), (I4-H4), 0)
Column K then adds displays the sub-total of Rules 1 and 2 on the one hand and Rule 3 on the other and reveals the fundamental flaw of the Second Formula:
=H4+J4
This sum in K13 adds up to 256 instead of 254. The Second Representation Formula should have stopped right here and separated the electoral divisor from the maximum number of provincial MPs in the House of Commons. This would have recognised the Senatorial Clause as an aberration over and above the principle of representation by population and separate from the electoral divisor. But since the Second Formula insists on integrating these two incompatible things, Prince Edward Island’s two extra MPs under the Senatorial Clause ultimately end up taking away one MP each from Ontario and Quebec. Under rule 3 alone, Ontario got 84 MPs and Quebec got 74; but by the end of the sheet, Ontario only came away with 83 and Quebec with 73
Rule 4 (Rule 1, 2nd Round)
Rule 4 is essentially Rule 1, Round 2 and appears in columns L, M, N, and O in light yellow. Since Prince Edward Island obtained an additional two MPs under Rule 3, we must now re-do the calculation under Rule 1 after removing Prince Edward Island’s population and number of MPs from the total provincial population in column L (=B13-B4) and the electoral divisor in column M (=C13-K4). These produce, in turn, a higher electoral quotient in column M (=L$13/M$13).
Column N then replicates column D and shows the new number of MPs per province before rounding down: (=B5/L5). And column O repeats column E to show the number of MPs rounded down: =ROUNDDOWN((N5),0). The sum of column O now only adds up to 247, which means that the fractional remainders add up to 3: =SUM(O5:O12).
Rule 2, 2nd Round
This stage takes up columns P, Q, and R, also in light blue. Once more, I applied a TRUNC and conditional function to column P to display the fractional remainders and then highlight the three largest amongst them in grey: =N5-TRUNC(N5)
In column Q, I then entered a “1” manually in the three rows with the three largest fractional remainders in column Q (which Calc identified using that same conditional formatting described earlier, but calibrated to the top 3 values in column P). Finally, column R lists the sub-total under Rule 4 and Rule 2, 2nd round – but only for the provinces to which Rule 4 applied – by adding the number of MPs rounded down in column O and the remainders redistribution in column R: =O5+Q5. Since this sum then adds up to 250, everything is accounted for: =SUM(R4:R12)
The Grand Total
Column S then lists the grand total number of MPs per province by adding up the sub-totals from Rule 3 and Rule 2, 2nd Round: =IF(R4=0, K4,R4)
The final sum then comes to 254 and matches the electoral divisor. These figures also match those under the Representation Act, 1947, shown for reference in column T.
The Nova Scotian Anomaly in the 1970s
I describe in the book how Andrew Sancton (Professor of Political Science at the University of Western Ontario) found that the Representation Commissioner had miscalculated the number of MPs per province under the Third Formula, and that this discrepancy between whether Ontario got 91 or 92 MPs arose based on whether Nova Scotia obtained 10 MPs under the Senatorial Clause (Rule 3) or from rounding and distributing the largest remainders (Rule 2). A similar Nova Scotian Anamoly crops up in the 2nd Formula in the 1970s and ultimately results in a House of Commons of 262 provincial MPs instead of 261 – which should be impossible under the 2nd Formula. This might show a gap in the rules themselves. But I ran through two configurations of running through rules 1 and 2 twice and rule 3 once, and running through rules 1 and 2 thrice and rule 3 twice, and they both resulted in the same conclusion: by whatever means Nova Scotia gets its minimum of 10 MPs, the House of Commons still ends up with 262 provincial MPs instead of 261 of them.
Nothing like this ever happened in the other decades, and it has left me puzzled. I can only presume that this error cropped out because of the vagueries of rounding. In Column s on the sheet “1970s (Short Method)”, I modified the if-then function into a compounded if-then – OR function as follows, because Nova Scotia just get at least 10 MPs:
=IF(OR((R4=0), (O4<I4)), K4,R4)
Then in the sheet “1970s (Long Method)”, where I repeated rules 1 and 2 for a third time and rule 3 for a second time, I reverted back the regular if-then function, but using different columns as references to reflect the extra stage in the calculation: =IF(AC4=0, L4,AC4)
Perhaps I made a mistake somewhere, but the total for every other decade came out right, at 261 provincial MPs.
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