General discussion about Congressional Apportionment

[Antonio the Sixth]

Since I had nothing else to do, I spent the past weeks of my life (when I wasn't on the forum ) to redo myself the entire 2003 congressional apportionment based on the 2000 Census Bureau's population results. That means I calculated the priority value for each single seat using the formula I found on wikipedia.

Here's what it says :

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To put it clearly, Priority Value=Population/(n*n+1)^0.5

I realized a list of every seat using this system, but it's only at the very end that I realized there was a problem. It obviously came to the very controversial NC-13 and UT-4 seats... And here is how it ends up :

North Carolina : 8,049,313 inh.
8049313/(12*13)^0.5=644461

Utah : 2,233,169 inh.
2233169/(3*4)^0.5=644660

On this count, UT-4's priority value is 199 higher than NC-13's. So, if these population counts are correct, Utah should have been granted the 435th seat.

Instead, if we look at the official census Priority values, we get :

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How can we explain this difference ?

[Joe Cooper]

Military is separately assigned to the states for purposes of apportionment.

[jimrtex]

Since I had nothing else to do, I spent the past weeks of my life (when I wasn't on the forum ) to redo myself the entire 2003 congressional apportionment based on the 2000 Census Bureau's population results. That means I calculated the priority value for each single seat using the formula I found on wikipedia.

Here's what it says :

You must be logged in to read this quote.

To put it clearly, Priority Value=Population/(n*n+1)^0.5

I realized a list of every seat using this system, but it's only at the very end that I realized there was a problem. It obviously came to the very controversial NC-13 and UT-4 seats... And here is how it ends up :

North Carolina : 8,049,313 inh.
8049313/(12*13)^0.5=644461

Utah : 2,233,169 inh.
2233169/(3*4)^0.5=644660

On this count, UT-4's priority value is 199 higher than NC-13's. So, if these population counts are correct, Utah should have been granted the 435th seat.

Instead, if we look at the official census Priority values, we get :

You must be logged in to read this quote.

How can we explain this difference ?

The "apportionment population" is the sum of the resident population and the overseas population.

The overseas population consists of federal government employees, military and civilian and dependents who live with them.  The courts have deferred to the judgment of Congress and the federal government about how the census and apportionment is conducted.  These are recent court decisions on the matter.

Franklin v Massachusetts

Following the 1990 census, Massachusetts had challenged the inclusion of the overseas population in the apportionment population.  This cost Massachusetts its 11th representative.  Because much of the overseas population included in the apportionment population is in the military or their dependents, this tends to have a bias towards states with military bases.  The military has a number of different states that can be attributed to military personnel.  If possible, they use state of residence when they entered the military, but for some people this information isn't available.

Most of the above case deal with legalities of how the Commerce Department and Census Bureau decided to include the overseas population, rather than whether the inclusion was lawful.

Department of Commerce v Montana

Following the 1990 census, Montana lost its 2nd representative, and sued, arguing that the wrong divisors were used in the determining the priority order.

Possible divisors for determining the priority value for the n+1 representative are:

((n)*(n+1))^0.5  what is currently used.

((n)+(n+1))/2  this is more favorable to large states.  It is equivalent to using St.Lague
divisors.  If this were used in 2010, Rhode Island might lose its 2nd seat.

2*(n)*(n+1) / (n)+(n+1) which is more favorable to smaller states.  If you were to repeat your exercise with these divisors, you would find that Montana would get a 2nd seat.  It might be that this would let Utah pass North Carolina, but not gain the 435th seat because Montana would steal it.

(n+1)  This is equivalent to D'Hondt and is very unfavorable to smaller states.

(n)      This is very favorable to small states.

Incidentally, the lawyers making the oral arguments were Marc Racicot, who was Attorney General of Montana at the time, later governor, and then chairman of the Republican National Committee in the early 2000's; and Ken Starr, special prosecutor in the Clinton corruption cases and recently appointed President of Baylor University.

Wisconsin v City of New York

This was about the use of sampling.

Utah v Evans

This was another case involving census methodology.  This was triggered by the determination that North Carolina had narrowly edged out Utah for the 435th seat. 

Utah had considered suing based on the census bureau not counting Mormon missionaries in the overseas, but eventually decided they didn't have a good case.  They did get the Census Bureau to consider other counting methods for US citizens who reside overseas, but are not affiliated with the federal government.  I suspect that if the federal government took a more direct role in the registration of US citizens resident overseas they could get a reasonably accurate count.  After all, the whole purpose of including them is for purposes of apportioning representatives and the federal government requires that they vote in congressional elections.

Anyhow, Utah discovered that if the census had used another method of conducting the resident census, Utah would have got the 435th seat.

The closeness of Utah to getting the 435th seat, is also why it was given the 437th seat under proposals that would have granted the District of Columbia congressional representation.

Muon's projections of apportionment for 2010 do include an adjustment for the overseas population.  The Census Bureau only makes estimates of the the resident population.  The overseas population is only tabulated for apportionment purposes - the Census Bureau does not publish any details other than statewide counts.

So Muon's projects are based on the resident population as estimated by the Census Bureau and then adjusts the population for each state based on an assumption that the ratio of overseas to resident population is constant for each state.

[Antonio the Sixth]

Thanks a lot for all these precisions, Jim.
One last thing : Do overseas residents vote for House elections in the State they are assigned to ? If not, counting them makes no sense.

And BTW, I think the arithmetic mean would be fairer as divisor than the geometric one. The current system clearly advantages small States (a State whose population is equivalent to 1.41 seats has as much chances to get its second seat as one of 52.5 to get its 53th). This is particularly senseless, considering that anyways small States get overrepresented thanks to the Senate.

[True Federalist (진정한 연방 주의자)]

I think the arithmetic mean would be fairer as divisor than the geometric one. The current system clearly advantages small States (a State whose population is equivalent to 1.41 seats has as much chances to get its second seat as one of 52.5 to get its 53th). This is particularly senseless, considering that anyways small States get overrepresented thanks to the Senate.

Depends on what one considers "fair".  Using the geometric mean keeps the range of people per representative smaller.

For sake of example, assume we have a population of 435 million to apportion among 435 seats for an average of 1,000,000 per seat.

Assuming no State is getting its first representative because of the floor of 1 seat, using the arithmetic mean gives a range in potential district sizes of 750,000 to 1,500,000 while the geometric mean gives a range of 707,107 to 1,414,214.  Hence the geometric mean gives a 5.72% smaller variance in district sizes than the arithmetic mean does.

Another method of reducing the variance is to increase the number of seats, and not just because of the smaller averages.  For example, if we had an average district size of 1,000,000 but each State had enough population to be guaranteed two districts, then AM has a range of 833,333 to 1,250,000 (416,667) and GM has a range of 816,497 to 1,224,745 (408,248) both of which are considerably smaller ranges than if having 1 seat is a possibility.  GM still has an advantage, but it is a smaller advantage (2.02%).  No matter how many seats is the minimum of honestly earned seats, GM will always be have the smaller variance in district sizes, tho the advantage will be smaller as the number of seats increases.

BTW, the reason I oppose statehood for the Virgin Islands, American Samoa, Guam and the Northern Mariana Islands is that they don't have enough population to warrant a Representative of their own without the floor.  (Even if Guam and the NMI were to unite as the Mariana Islands, they'd be too small population-wise.)

[Antonio the Sixth]

Yes, I see, it always comes to the opposition between those who think "fairness" means the smaller differences in terms of people per seat and those who think it means the  smaller differences between the theorical number of seats and the actual number. I guess it's gonna be an endless debate.

Anyways, the best solution would be indeed to dramatically raise the number of seats in the House. The US should have at least 650, and even then they would be far from other democracies in terms of representativity.

[True Federalist (진정한 연방 주의자)]

I'm in favor of the cube root rule myself, which by the Census Bureau's estimates would give the US House 676 seats. and UN if it had a body apportioned by population 1,895 seats.

[Bo]

Nice job Antonio. I know that I would personally never bother spending so much time on something like that.

[Obnoxiously Slutty Girly Girl]

Nice job Antonio. I know that I would personally never bother spending so much time on something like that.

LOL, wouldn't want to take any time away from your busy lifestyle of posting here 79 posts a day....

[Хahar 🤔]

I swear, Jim's a robot.

[jimrtex]

One last thing : Do overseas residents vote for House elections in the State they are assigned to ? If not, counting them makes no sense.

No.

The overseas population included in the apportionment count are only federal employees (military and civilian).  They are not enumerated in the manner of US residents, who are sent a census form that they fill out and return.  Instead they are counted based on administrative records of the agency they are affiliated with.  In 1990, 98% of the overseas population was associated with the Department of Defense (DOD), both military and civilian, plus dependents who are living with them.

1990 Overseas Population see pg 62

In the case of the DOD military personnel, the state is generally the "Home of Record", which is where the person lived when they entered the service (or in some cases, re-enlisted).  If this isn't available, then the state indicated on their records for tax withholding purposes.  Some states in the US don't have state income taxes, so many military personnel claim that state as their residence for tax purposes.  Other possibilities include US location for last assignment of 6 months or longer, or last US assignment.

Overseas resident voters may vote in their place of last residence in the United States.  There really isn't any government records of all these persons.  Registering with an embassy or consulate is voluntary.  US citizens are generally free to exit and enter the US.  There might be as many 6 million.

Issues of Counting Americans Overseas in Future Censuses

Whether some are actually resident overseas is vague.  Someone who retires may keep their house in the US, and visit it for a few months a year.  It may be easier to keep their voting residence in the US, and then vote absentee where they live most of the year.

There is a mail forwarding service in Texas that maintains permanent mailing addresses for RV'ers.  So that it doesn't look like a PO box, they use "street names".  Some people have registered to vote based on that address, and it has been an ongoing controversy whether they were resulting in Republicans taking control of the county.

And BTW, I think the arithmetic mean would be fairer as divisor than the geometric one. The current system clearly advantages small States (a State whose population is equivalent to 1.41 seats has as much chances to get its second seat as one of 52.5 to get its 53th). This is particularly senseless, considering that anyways small States get overrepresented thanks to the Senate.

The arithmetic mean minimizes the deviation in the number of representatives per person.  If there was one representative per 1 million persons (for the country).  Then a state that was right on the threshold between having one and two representatives, of having 2/3 as many representatives per person as the ideal, to having 4/3 as many representatives per person.

The harmonic mean minimized the deviation in the number of persons per representative (district size).  A state that was right on the threshold between one and two representatives would go from having a single district that was 4/3 the ideal size to one that was 2/3 the ideal size.  This is what Montana was arguing for.

The geometric mean is somewhere in between.

arithmetic_mean/geometric_mean = geometric_mean/harmonic_mean

[Antonio the Sixth]

Ok, so if they are able to vote in their State of origin, the inclusion of overseaers is legitimate. Obviously, all that would be easily solved if a "americans overseas" district (or several, according to their number : 6 milions would give them 9 seats) were created.

I didn't know the existence of the harmonic mean, and I've no idea what it is. Could you explain me better what it consists in ? Thanks.

[muon2]

Ok, so if they are able to vote in their State of origin, the inclusion of overseaers is legitimate. Obviously, all that would be easily solved if a "americans overseas" district (or several, according to their number : 6 milions would give them 9 seats) were created.

I didn't know the existence of the harmonic mean, and I've no idea what it is. Could you explain me better what it consists in ? Thanks.

The harmonic mean was popular with the ancient Greeks, since it had nice geometric properties. In modern use it might come up in a problem like this:

Antonio walks from his house to the post office at a speed of 2 km/hour and returns at a speed of 3 km/hr. What is his average speed during the round trip?

The answer is not 2.5 km/hr, but instead it is the harmonic mean of 2 and 3 which is 12/5= 2.4 km/hr. If the post office were 3 km away it would take 1.5 hours to get there and 1 hour to return at total of 2.5 hours. The round trip is 6 km, so the average speed is 6 km/2.5 hr = 2.4 km/hr.

[memphis]

I think the arithmetic mean would be fairer as divisor than the geometric one. The current system clearly advantages small States (a State whose population is equivalent to 1.41 seats has as much chances to get its second seat as one of 52.5 to get its 53th). This is particularly senseless, considering that anyways small States get overrepresented thanks to the Senate.

Depends on what one considers "fair".  Using the geometric mean keeps the range of people per representative smaller.

For sake of example, assume we have a population of 435 million to apportion among 435 seats for an average of 1,000,000 per seat.

Assuming no State is getting its first representative because of the floor of 1 seat, using the arithmetic mean gives a range in potential district sizes of 750,000 to 1,500,000 while the geometric mean gives a range of 707,107 to 1,414,214.  Hence the geometric mean gives a 5.72% smaller variance in district sizes than the arithmetic mean does.

Another method of reducing the variance is to increase the number of seats, and not just because of the smaller averages.  For example, if we had an average district size of 1,000,000 but each State had enough population to be guaranteed two districts, then AM has a range of 833,333 to 1,250,000 (416,667) and GM has a range of 816,497 to 1,224,745 (408,248) both of which are considerably smaller ranges than if having 1 seat is a possibility.  GM still has an advantage, but it is a smaller advantage (2.02%).  No matter how many seats is the minimum of honestly earned seats, GM will always be have the smaller variance in district sizes, tho the advantage will be smaller as the number of seats increases.

BTW, the reason I oppose statehood for the Virgin Islands, American Samoa, Guam and the Northern Mariana Islands is that they don't have enough population to warrant a Representative of their own without the floor.  (Even if Guam and the NMI were to unite as the Mariana Islands, they'd be too small population-wise.)

You oppose statehood for Wyoming also? It'd be a lot fairer (though also completely absurd) if we could gerrymander a couple hundred thousand Montana voters into Wyoming's district.

[Antonio the Sixth]

Ok, so if they are able to vote in their State of origin, the inclusion of overseaers is legitimate. Obviously, all that would be easily solved if a "americans overseas" district (or several, according to their number : 6 milions would give them 9 seats) were created.

I didn't know the existence of the harmonic mean, and I've no idea what it is. Could you explain me better what it consists in ? Thanks.

The harmonic mean was popular with the ancient Greeks, since it had nice geometric properties. In modern use it might come up in a problem like this:

Antonio walks from his house to the post office at a speed of 2 km/hour and returns at a speed of 3 km/hr. What is his average speed during the round trip?

The answer is not 2.5 km/hr, but instead it is the harmonic mean of 2 and 3 which is 12/5= 2.4 km/hr. If the post office were 3 km away it would take 1.5 hours to get there and 1 hour to return at total of 2.5 hours. The round trip is 6 km, so the average speed is 6 km/2.5 hr = 2.4 km/hr.

Oh, that's weird but interesting.
So if I well understand the HM's formula is (n^2*(n+1))/(n+(n+1)) ?

[Kevinstat]

Ok, so if they are able to vote in their State of origin, the inclusion of overseaers is legitimate. Obviously, all that would be easily solved if a "americans overseas" district (or several, according to their number : 6 milions would give them 9 seats) were created.

I didn't know the existence of the harmonic mean, and I've no idea what it is. Could you explain me better what it consists in ? Thanks.

The harmonic mean was popular with the ancient Greeks, since it had nice geometric properties. In modern use it might come up in a problem like this:

Antonio walks from his house to the post office at a speed of 2 km/hour and returns at a speed of 3 km/hr. What is his average speed during the round trip?

The answer is not 2.5 km/hr, but instead it is the harmonic mean of 2 and 3 which is 12/5= 2.4 km/hr. If the post office were 3 km away it would take 1.5 hours to get there and 1 hour to return at total of 2.5 hours. The round trip is 6 km, so the average speed is 6 km/2.5 hr = 2.4 km/hr.

Oh, that's weird but interesting.
So if I well understand the HM's formula is (n^2*(n+1))/(n+(n+1)) ?

No, it's (2*n*(n+1))/(n+(n+1)), or 2n(n+1)/(n+(n+1)), or (2n^2+2n)/(2n+1)) as mathematicians might write it, but I prefer the middle one.

The harmonic mean of two numbers (or any number of numbers) is the reciprocal of the arithmetic mean of the reciprocal of those numbers.  Following order of operations, with multiplication and division ahead of addition in the absence of parentheses (exponents, which trump multiplication and division, and subtraction which is equal in priority to addition are not included here), ...

1/(((1/n+1/(n+1)))/2) = 2/((1/n+1/(n+1))) =  2/(((n+1)+n)/(n(n+1))) = 2n(n+1)/(n+(n+1)) .  This is for the "n+1th" seat.  The harmonic mean formula for a priority value the nth seat would be 2n(n-1)/(n+(n-1)) .

[Antonio the Sixth]

Oh yeah, I see now.
(1/3+1/2)/2=0.416667
1/0.416667=2.4

Interestingy, the geomentric mean between two numbers is also the geometric mean between their arithmetic mean and their geometric mean...

[True Federalist (진정한 연방 주의자)]

BTW, the reason I oppose statehood for the Virgin Islands, American Samoa, Guam and the Northern Mariana Islands is that they don't have enough population to warrant a Representative of their own without the floor.  (Even if Guam and the NMI were to unite as the Mariana Islands, they'd be too small population-wise.)

You oppose statehood for Wyoming also? It'd be a lot fairer (though also completely absurd) if we could gerrymander a couple hundred thousand Montana voters into Wyoming's district.

No.  If we added one more seat to the HoR and doubled Wyoming's population in 2000, it would have legitimately earned 2 seats under the current apportionment formula (its 2nd seat would be the 387th seat under that scenario), so there's no reason for Wyoming to not be a State.  For the 2000 Census the cutoff point for Statehood under my criteria would be 456,567 (half of the minimum population needed to get a second Representative in a House with 436 members). Wyoming was above  that with an apportionment population 495,304.

The population of all four minor territories combined in 2000 was only 389,929 so even together they didn't meet the target. Possibly the combined Marianas (NMI + Guam) might be able to grow into Statehood, especially if we ever increased the size of the House to use the cube root formula as both Guam and NMI are growing at a faster rate than the U.S. With 676 seats in 2010, the target number would be approximately 322 thousand and the combined Marianas will have approximately 270 thousand.  (Kept at 435 seats and the target number for 2010 will be approximately 500 thousand, and Wyoming will again be comfortably over that number.)

[jimrtex]

Ok, so if they are able to vote in their State of origin, the inclusion of overseaers is legitimate. Obviously, all that would be easily solved if a "americans overseas" district (or several, according to their number : 6 milions would give them 9 seats) were created.

I didn't know the existence of the harmonic mean, and I've no idea what it is. Could you explain me better what it consists in ? Thanks.

Harmonic mean is:  2*(n)*(n+1) / (n)+(n+1)

4/3
12/5
24/7
40/9

or alternatively:

1¹/₃
2²/₅
3³/₇
4⁴/₉

The arithmetic, geometric, and harmonic mean all converge on n¹/₂

If you use the harmonic mean, you minimize the variation in district size.

Let's say that a state has a population that is somewhere between 1 and 2 times the ideal district population.

If the harmonic mean is used, when its population is just below 1.33 the ideal, it will have a single district with a population 33% larger than the ideal.  Increase the population a bit, and it will have two districts whose population is 33% smaller than the ideal.

A similar thing happens around 2.4 times the ideal, when it goes from 2 districts that are 20% too large, to 3 districts that are 20% too small.

If you use the arithmetic mean, a single district would be 50% too large, and then two districts 25% too small.   Or two districts that are 25% too large, and then three districts that are 17% too small.

[jimrtex]

Ok, so if they are able to vote in their State of origin, the inclusion of overseaers is legitimate. Obviously, all that would be easily solved if a "americans overseas" district (or several, according to their number : 6 milions would give them 9 seats) were created.

I didn't know the existence of the harmonic mean, and I've no idea what it is. Could you explain me better what it consists in ? Thanks.

The harmonic mean was popular with the ancient Greeks, since it had nice geometric properties. In modern use it might come up in a problem like this:

Antonio walks from his house to the post office at a speed of 2 km/hour and returns at a speed of 3 km/hr. What is his average speed during the round trip?

The answer is not 2.5 km/hr, but instead it is the harmonic mean of 2 and 3 which is 12/5= 2.4 km/hr. If the post office were 3 km away it would take 1.5 hours to get there and 1 hour to return at total of 2.5 hours. The round trip is 6 km, so the average speed is 6 km/2.5 hr = 2.4 km/hr.

Or, Antonio lives in a country which apportions one representative in its Congress per 100,000 people.  His state has 240,000 persons or the equivalent of 2.4 representatives.  So the question is whether his state should have 2 or 3 representatives.  And it so happens that 2.4 is the harmonic mean of 2 and 3.

If Antonio was a congressman, and the state was apportioned 2 seats, he would represent 120,000 persons or 20,000 (20%) more than the ideal.  If the state was apportioned 3 seats, he would represent 80,000 persons or 20,000 (20%) less than the ideal.  It could be considered fair to apportion the state either 2 or 3 representatives, since the deviation in the district size from the ideal would be the same in either case.  If Antonio's state had 1000 more people, then  the deviation for 2 representatives would be +20,500 but for 3 representatives would be -19,667.  So it would be better to apportion the 3rd representative since it reduces the deviation from the ideal.  And if the population were slightly less than 240,000, the opposite would be true.  The deviation would be less if there were two representatives.

On the other hand, perhaps fairness should not be determined based on persons per representative, but rather representatives per person.  Don't we want to let each person have about the same chance of influencing the decision of his country through his representative.

It will be easier to see if instead of using representatives / person, we use representatives per 100,000 persons.  The result is the same, but instead of small fractions (0.00001) we end up with numbers closer to 1.

If Antonio is no longer a representative, but just an ordinary citizen, then if his state of 240,000 persons is apportioned 2 representatives, there will 0.833 representatives per 100,000 persons.  If there are 3 representatives, there will be 1.250 representative per 100,000 persons.  In one case, Antonio will receive 16.7% less than the ideal representation, while in the other, he will receive 25.0% more representation.  We can't give him the ideal amount of representation, but we can give him as close to the ideal as possible by apportioning his state 2 representatives.

It would not be until his state grew to 250,000 persons (2.5 times the ideal, and the arithmetic mean of 2 and 3) that it would be fairer in terms of representation to apportion a 3rd seat.  If there are 3 representatives for the 250,000 persons, there will be 1.20  representatives per 100,000; and if there are 2 representatives there will be 0.80 representatives per 100,000.  So the over-representation on the one hand and under-representation on the other are balanced at 20%.

[True Federalist (진정한 연방 주의자)]

As for the geometric mean that jimrtex omitted in in his previous post, it's justification is as follows :

If the population were the geometric mean of 200,000 and 300,000 (244,949 if rounded to the nearest integer) then if you had two representatives you would have 100,000*sqrt(1.5) people per representative and if you had three representatives you would have  100,000/sqrt(1.5) people per representative.

(Conversely with two representatives you have 1/sqrt(1.5) per 100,000 people and with three you have sqrt(1.5) per 100,000 people.)

Hence by the arguments jimrtex put forth concerning whether people per representative or representatives per people is the criterium to judge by, the geometric mean is equally fair to both criteria.  (What using the geometric mean minimizes is the difference in the logarithm of both ratios from the ideal value.)

The geometric mean, m,  of any two numbers a and b is the number such that m/a = b/m, which is why the multiplier and the divisor are the same above.  Insert n for a and n+1 for b and then solve for m and you get m=sqrt(n*(n+1)).

The geometric mean above is thus 100,000*sqrt(6).

100,000*sqrt(6)/2=100,000*sqrt(6)/sqrt(4)=100,000*sqrt(6/4)=100,000*sqrt(1.5)
100,000*sqrt(6)/3=100,000*sqrt(6)/sqrt(9)=100,000*sqrt(6/9)=100,000*sqrt(9/6)=100,000/sqrt(1.5)


The arithmetic mean does have one advantage over the geometric and harmonic means, which, while not relevant to apportionment where every contender for seats is guaranteed at least one seat, pertains to allotting seats in a proportional representation election.  Namely that that there is a natural cutoff between having 0 seats and having 1 seat.  This is because both the harmonic and geometric mean of 0 and any other number is 0, so if an unmodified geometric or harmonic mean were used to allocate seats, every party that got at least one vote would be guaranteed 1 seat.

[Kevinstat]

BTW, the reason I oppose statehood for the Virgin Islands, American Samoa, Guam and the Northern Mariana Islands is that they don't have enough population to warrant a Representative of their own without the floor.  (Even if Guam and the NMI were to unite as the Mariana Islands, they'd be too small population-wise.)

You oppose statehood for Wyoming also? It'd be a lot fairer (though also completely absurd) if we could gerrymander a couple hundred thousand Montana voters into Wyoming's district.

No.  If we added one more seat to the HoR and doubled Wyoming's population in 2000, it would have legitimately earned 2 seats under the current apportionment formula (its 2nd seat would be the 387th seat under that scenario), so there's no reason for Wyoming to not be a State.  For the 2000 Census the cutoff point for Statehood under my criteria would be 456,567 (half of the minimum population needed to get a second Representative in a House with 436 members). Wyoming was above  that with an apportionment population 495,304.

The population of all four minor territories combined in 2000 was only 389,929 so even together they didn't meet the target. Possibly the combined Marianas (NMI + Guam) might be able to grow into Statehood, especially if we ever increased the size of the House to use the cube root formula as both Guam and NMI are growing at a faster rate than the U.S. With 676 seats in 2010, the target number would be approximately 322 thousand and the combined Marianas will have approximately 270 thousand.  (Kept at 435 seats and the target number for 2010 will be approximately 500 thousand, and Wyoming will again be comfortably over that number.)

Well, under the equal proportions (geometric mean), harmonic mean or smaller divisors methods, the divisor for a state to get its first seat, susing the same formula as for all additional seats under the respective methods, would be 0.  The limit as n approaches 0 from the positive side of any positive number/n is (positive) infinity, so under either of those three methods (including the current equal proportions method), the priority value for the 1st seat of any state with at least one resident or overseas person attibuted to that state would be infinity and the largest s states would have at least one seat up to the number s of seats, with any remaining seats being awarded on the basis of the non-infinite priority values.  That's the one thing I like about the equal proportions method as opposted to major fractions, besides the fact that my state could potentially benefit from it in a couple decades or so.  The constitutional requirement that each state, regardless of population, have at least one seat goes naturally with it.

Your doubling of a states population to see if it would get two seats seems disjoint to me given how the equal proportions method works.  The divisor for a state's second seat is less than half the divisor for a state's forth seat in any of the main divisor methods detailed by jimrtex except for the greatest divisors method, so why should we assume the divisor for a state's first seat if there was no "floor" or 1-seat guarantee would be half the divisor for a state's second seat under any of those same methods?

In Wyoming's case it would probably get a seat under major fractions with no 1-seat guarantee (the divisor for a state's first seat in that method would be .5, so you would multiply a state's apportionment population by 2 to get its priority value for its first seat), but probably not under greatest divisors with no such guarantee (where the divisor for a state's first seat would be 1), which is the only one of the five main divisor methods (with or without a 1-seat guarantee) where the divisor for a state's nth seat is exactly half the divisor for a state's 2nth seat.

Wow, True Federalist beat me to the point I was trying to make, not as well stated as he did, about the equal proportions (geometric mean) method not having a natural cutoff between 0 seats and having 1 seat, but I see that as an advantage to the equal proportions method as far as seat allocation among U.S. states is concerned because it goes with the U.S. Constitution's guarantee of at least 1 seat in the House of Representatives for each state.

[Kevinstat]

Also, Nevada wouldn't have had a single representative in the U.S. House under the methods used at the time with no floor for much of its time as a state, stretching through 1930s reapportionment and I believe through the 1940s reaportionment if the Congressional Dems hadn't switched from major fractions to equal proportions in 1941 to prevent then-solidly Democratic Arkansas from losing a seat to then-Republican-leaning Michigan.

[True Federalist (진정한 연방 주의자)]

Also, Nevada wouldn't have had a single representative in the U.S. House under the methods used at the time with no floor for much of its time as a state, stretching through 1930s reapportionment and I believe through the 1940s reaportionment if the Congressional Dems hadn't switched from major fractions to equal proportions in 1941 to prevent then-solidly Democratic Arkansas from losing a seat to then-Republican-leaning Michigan.

True, but Nevada was granted Statehood way too early so as to gin up three Union electoral votes in 1864.  Had it not been for concerns over the Confederacy and the Mormons, it would have been better to have split Nevada between Utah and California and then split California into a North and South California.  Of course had that happened, there never would have been a Las Vegas.

[muon2]

Also, Nevada wouldn't have had a single representative in the U.S. House under the methods used at the time with no floor for much of its time as a state, stretching through 1930s reapportionment and I believe through the 1940s reaportionment if the Congressional Dems hadn't switched from major fractions to equal proportions in 1941 to prevent then-solidly Democratic Arkansas from losing a seat to then-Republican-leaning Michigan.

True, but Nevada was granted Statehood way too early so as to gin up three Union electoral votes in 1864.  Had it not been for concerns over the Confederacy and the Mormons, it would have been better to have split Nevada between Utah and California and then split California into a North and South California.  Of course had that happened, there never would have been a Las Vegas.

Las Vegas might have happened anyway. Without the state of Nevada, the Clark county area was part of New Mexico territory. It wasn't even part of NV until 2 years after statehood. This is an 1857 map of NM terr: