The DK Index put forward by Dauer and Kelsay applied to the least number of votes needed to take a majority of seats in HofC

DK Index

named after Dauer and Kelsay, 1955

evaluates the smallest size of a population of citizens which control a majority of representatives in the assembly.

usually based on geographical representation 

so taking Canada, the total population of the 172 smallest districts.

(as if the district was truly represented by the the single member in each district and as if that was the worst of the mis-representation occurring under FPTP)

But the DK Index could be made to work in a more telling fashion -- by looking at the least-popular 172 MPs, the elected members who received the least number of votes.

Their vote total would be a truly scary number in its smallness - perhaps just 10 percent.

Perhaps someone who worked out effective votes in the last election has info on the votes taken by each successful candidate.

or we could derive the theoretically possible least-vote majority - 

with say 22 percent of the vote in the least-populous districts - 

could be as few as ten percent or so.

spitballing:

100,000 pop. on average in a district

66,000 voters in a district

66 percent voter turnout  = 44,000 votes cast in a district

22 percent is 10,000 votes per district

172 MPs makes up a majority

so perhaps

only 1.7M votes would elect a majority of MPs in the House of Commons,

out of the 19M votes cast by the 40M people in Canada.

Truly scary!

footnote:  

DK index discussed here

wp_tse_1089.pdf

there was another place online with more info, applying it to the U.S. but can't find it now.

see

A Measure of the Population Quality of Legislative Apportionment*

Published online by Cambridge University Press:  01 August 2014

Henry F. Kaiser

American Political Science Review , Volume 62 , Issue 1 , March 1968 , pp. 208 - 215

DOI: https://doi.org/10.2307/1953335[Opens in a new window]

AND

Dauer, Manning J. and Kelsay, Robert G., “Unrepresentative States,” National Municipal Review, 44 (1955), 515–575, 587.

"Unrepresentative States

From 1937 to 1955 legislatures’ representativeness declined in more than three-quarters of the states."

By MANNING J. DAUER and ROBERT G. KELSAY*

https://archive.org/details/sim_national-civic-review_1955-12_44_11/page/570/mode/2up

...

The basic assumption of the study is that a legislature which does not equably represent the people of the state contains an innate weakness which limits the effective functioning of that government on many types of problems. It is assumed further that a measure of the representative character of the legislatures can be achieved by determining the smallest percentage of a state’s population which could theoretically elect a majority of the lower house and a corresponding figure for the upper house under present district boundaries.

...

Certain generalized observations can be made as to why a state is high or low in the listings.

The worst situations exist in those states where a constitutional or statutory limitation sets a rigid system which recognizes some unit of local government as the basis of representation, such as the town or county. Connecticut, for example, elects at least one representative from each town, while in Nevada one senator is elected from each county.

Along with using a local unit as an area of representation, some states ranking low establish rigid representation ratios. In these instances no one unit can have more than a certain number of legislators, thereby disregarding the relative difference in size between the largest and smallest units. Thus the Florida constitution establishes a ratio of three to one between the largest and smallest counties for representation in the lower house, although the counties range in population from nearly 500,000 down to about 2,000.

Several variations of such rigid limitations could be mentioned, but invariably they have the effect of giving sparsely populated areas considerable over-representation.

[Representative states:]

Many of the states near the top of the rankings are found there because of frequent legislative action in making changes in district boundaries in order to adjust to population movements. In some instances this action has been taken more or less on a voluntary basis. In still others, some kind of constitutional self-executing provision has given the impetus necessary to insure action by bringing non-legislative agencies into the apportionment picture. Under a Missouri constitutional provision, a non-legislative body has the initial responsibility for reapportionment, while in the new Illinois amendment, a non-legislative commission is used only if the legislature fails to act.

Another type of provision that has tended to promote more equitable representation permits apportionment plans to be challenged in the courts, as in New York, Arkansas and some other states.

[Conclusion]

The effect of apportionment on the representativeness of our state legislatures has been pictured. The problems resulting from this have been discussed elsewhere. But the new data presented here should aid in determining where effort must be placed on apportionment.

========================

National Municipal Review 1958

https://archive.org/details/nationalmunicipa41natirich