Census Projections - New England States

[muon2]

In the next installment of my 2010 apportionment project, I've added the New England states to the links from the 2010 apportionment page. As before, comments or questions are welcome.

[CARLHAYDEN]

If the projections hold up, and I think they will, then the new england states will have lost eleven representatives in a century.

[muon2]

RI is the New England state that is most at risk for the next loss. They could lose in 2020, but 2030 is more likely with current projections.  CT and MA could also lose again in 2030 with current Census projections. Those projections have MA falling behind MD by 2030.

[Cubby]

Wow the New England info is depressing. The region hasn't gained a seat since 1930 (CT).

We should have gained one in 1970, after 3 decades of growth above the national average. If you divide CT's population that year (3,032,000) into the National total (203,302,000) and multiply it by 435, its 6.49. I'm not sure why we were cheated, except it may have to do with our 1 "at-large" seat, that I think was ruled unconstitutional in the 60's.

Mansfield, CT is the town where U-Conn is located. Its every Connecticut senior's safety school.

People care more about the U-Conn Huskies and other teams from that school than they do about any other sport, except maybe the Yankees/Mets & Jets/Giants.

[jimrtex]

We should have gained one in 1970, after 3 decades of growth above the national average. If you divide CT's population that year (3,032,000) into the National total (203,302,000) and multiply it by 435, its 6.49. I'm not sure why we were cheated, except it may have to do with our 1 "at-large" seat, that I think was ruled unconstitutional in the 60's.

If you simply rounded each states entitlement to the nearest whole number, you could have more or less than 435 representatives.

For example imagine if you had five states entitled to:

6.4, 5.4, 4.4, 3.4, and 2.4 representatives (collectively 22.0), if you round each state's representation independently, you end up with only 20 representatives.

But people leave the large state and migrate to the smaller states, you would have enititlements of:

5.6, 5.6. 4.6, 3.6, and 2.6, still 22.0 collectively.  But if you round each state's representation independently, you end up with 24 representatives.

So if you want to have exactly 435 representatives, you have to determine which states get an extra fraction of a representative and which don't.

The way this is done is by comparing how much each state is above the midpoint for its next seat, or how close it is to the midpoint, and ranking the states.  The highest ranked states are given the extra representative until 435 are apportioned.

The comparison is done on a relative basis.   So you might compare 6.49/6.50 for Connecticut, with 52.45/52.5 for California.  In such a case, California is actually higher ranked (0.9981 vs. 0.9990).

And secondly, the divisors are not the arithmetic mean (n + (n+1))/2 or n+1/2, but the geometric mean:  sqrt(n*(n+1)).   So the divisor for Connecticut's 7th seat is sqrt(6*7) or 6.481, while the divisor for Montana's 2nd seat is sqrt(1*2) or 1.414.

If you go through all the calculations, you will find that Connecticut was ranked 23rd among the states in their quest to get a favorable rounding.  Since 412 seats were given out based on the whole number of seats, or a minimum guarantee of 1 seat, which would mean Connecticut's 7th seat would have been 435th.

The reason Connecticut didn't get the 7th seat was that since 1970, apportionment of representatives has been based on what is called the "Apportionment Population" which includes overseas Federal civilian and military employees and their dependents.  The state that did get the 435th seat was Oklahoma, which was ranked 437th on the basis of resident population, but edged ahead of not only Connecticut, but Oregon on the basis of the overseas population.

The average population per representative based on the resident population was 465,468; while based on the apportionment population it was 469,088.  But this was not distributed proportionately throughout the states.  Oklahoma had about 26,000 in overseas apportionment population, vs. about 18,000 for Connecticut (and 19,000 for Oregon).

Apportionment Population

Based on the apportionment population, gained the 435th seat, Oregon was still ranked 436th, and Connecticut was 437th.

By the way, I found the following interesting article on apportionment methods.

Apportionment paradoxes

[muon2]

The apportionment paradox article seems to suggest a politically expedient way to avoid paradoxes and obey the quota rule. The best method would be to take the current number of seats and test to see if that number provides a match to the sum of seats based on the required rounding rule. If it does not,  apportion a number of seats equal to the nearest number above the current number which provides a match.

Using the 2000 census and the Huntington-Hill method the House would have 436 seats with that rule. Think of the money saved in Utah since they wouldn't have had to sue to try to get an extra seat.

[jimrtex]

The apportionment paradox article seems to suggest a politically expedient way to avoid paradoxes and obey the quota rule. The best method would be to take the current number of seats and test to see if that number provides a match to the sum of seats based on the required rounding rule. If it does not,  apportion a number of seats equal to the nearest number above the current number which provides a match.

Using the 2000 census and the Huntington-Hill method the House would have 436 seats with that rule. Think of the money saved in Utah since they wouldn't have had to sue to try to get an extra seat.

I had thought of a similar method, so I was interested that someone with a stronger mathematical background would come to the same conclusion.

I'm not sure that you should always apportion a number of representatives greater than or equal to the nominal number.   Imagine a case where the apportionment using independent rounding did yield a 435 member house, but there was one state, Unlucky, that was just short of the rounding threshold.

Now shift population from one state, Rusty, that was just above the rounding threshold, so that it is somewhat below the rounding threshold, to another state, Sunny, that was also above the threshold (the shift will be small enough that Sunny wll will have the same number of whole representatives, but large enough so that Unlucky is much closer to rounding threshold than Rusty.

So we apply our apportionment based on 435, and discover that we will only have 434 representatives.   So we try 436 and Unlucky now gets an additional representative but the total number is only 435.  So let's try 437, and now Rusty gets its representative back for a total of 437.   

It seems odd that Unlucky should get an additional representative, albeit in a slightly larger body, even though its share of the total population remained the same.  There would also be a ratcheting effect on the overall size of the house, since if we didn't start at 437, the next apportionment could cause more losses than gains, which was why we didn't accept the apportionment of 434 members in the first place.

If we just accept that certain states share of the total population didn't warrant an additional representative, then we could start from 435, and end up with a house likely to be within a member or 2 of 435, either way.  It should average close to 435 - there may be a slight upward bias due to a few small states (WY, ND, VT) permanently being rounded upward.  They will sometime be joined by SD.

So in 2000, instead of having decided whether NC or UT got the 435th seat, would have been to take away NC's 435th seat, and CA's 434th seat.

I did my application of Huntington-Hill in a slightly different manner.  Rather than using the quota (state-population/national-average) and rounding using modified thresholds (sqrt1*2), sqrt(2*3), etc.); I calculated a quota as:

  sqrt ( (state-population^2/national-average^2)  +1/4 )

And then used conventional rounding (fractions over 1/2 round upward)

If you sum these quotas, they come out to 436.84 which reflects the slight small state bias of Huntington-Hill.   In Bogomolny's article, if you compare the average value of S-N for values of N in the range 425 to 445, the average for Webster is -0.57, and for Huntington-Hill 0.33.

I would assume that over time, the fractional portion of the state quotas under Webster would be close to uniformally distributed between 0 and 1, and that S-N would average around 0 under Webster, and average 1.84 for Huntington-Hill (assuming little overall change in the distribution of population among large and small states).

I also adjusted the "average" used in Huntington-Hill so that the sum of quotas is 435 (this was done recursively, but convergence is very fast).  In the range around 435, this will increase slightly the chance of having a smaller house (S-N is negative).  But this may only be a random phenomena.  If you go further out to N=415 or N=455, S-N is positive, sometimes as much as +3.