Census Estimates for 2009 -> 2010 Apportionment

[jimrtex]

How close is RI to losing its second seat and MT and DE to gaining one?

RI will be just below a population equivalent to 1.5 seats, but because of the use of the harmonic mean, they only need about 1.41 x as much to keep 2nd seat.  Very likely will lose the 2nd seat in 2020 if it continues no growth pattern (the one district will be well over 1 million in population).

MT is just sputtering around just below the limit.  It was really close in 1990, had probably gained it back in the early 1990s and lost it, and just can't quite gain it back.  Lack of major cities prevents it from having the type of growth that Utah, Idaho, Colorado, Arizona have.  People like the mountains, but they need jobs.

DE is slowly gaining, but at the current rate it would be around 2050.

WV will quite likely lose 3rd seat in 2020, NE could.  ID could gain 3rd.

[jimrtex]

The following is the number of districts that each state would be entitled to based on its relative share of the total apportionment population, assuming that the square root divisors would still be used.  These are based on exponentially projecting the census estimate for July 2009 forward to April 2010.  There is no adjustment for the overseas population.

The second column is the change from the equivalent figure for 2000, so Wisconsin in essence has lost about 3/10 of a district between 2000 and 2010.  Wisconsin which lost its 9th seat in 2000, might expect its next loss in 2030.  

The 3rd column is the expected number of representatives.  What is remarkable is the number of states that may gain an extra seat, based not on their share of the total USA population, but simply because 435 representatives must be apportioned (it is akin to the situation of polling percentages not adding up to 100 because of rounding, but rather than simply inserting a footnote that the number of representatives do not add up to 435 due to rounding, the apportionment is adjusted so that it does.  

When the raw apportionment is short, as it is in this case it favors larger states because they can spread their deficit among more districts. So while California is about 183,000 people short of that needed to qualify for 52.5 representatives, this can be distributed so that each district is only about 3,500 under.  Meanwhile Montana's deficit is much smaller, but would have to be distributed between only two districts.

The 4th column is the projected percentage gain from 2000 to 2010.  States gaining faster than the national average of 9.9% are gaining in population share.  California is barely gaining faster than the national average, and there is a possibility that it will gain in population share while losing in representational share.

The 5th column is the percentage increase in the total population for a state to gain
(of its total population) for it to have one more seat, or to lose for it to lose its last seat.   The last two columns can be added to get an approximation of the change during the decade for
a different outcome to occur.  (eg if Wisconsin had lost 0.8% during the decade rather than gaining 5.9%, it would be faced with a loss of its 8th representative).

Most notable are the 7 states ranked around 435 whose quotients (used for ranking) are with 0.27% of one another.  Using ranking alone, one might envision a cross-country race where the runners cross the finish line as the race judge announces, "431 .... 432 433 ... 434 435 436 ... 437" etc.  Instead we have 7 states (AZ, MO, WA, CA, FL, TX, MN) finishing in a virtual dead heat, with NY and SC just a stride or two ahead, and OR and NC just behind.

The expected separation between the quotients for the 435th and 436th ranking would be the difference between the total population divided by 435 and the total population divided by 436, or about 1627.  The seven states are within 1938 of each other.  If we considered California to have a rank of 435, and gave the other states a (relative) ranking based on the relative difference in quotients, then they would rank:


NY 432.95
SC 433.70
AZ 434/70
MO 434.72
WA 434.74
CA 435.00
FL 435.57
TX 435.65
MN 435.89
OR 437.62
NC 438.09


Wisconsin             7.986   -0.302   8    5.9%  -6.7%
Connecticut           4.976   -0.300   5    3.6% -10.2%
New Hampshire         1.935   -0.035   2    7.8% -24.8%
Tennessee             8.923    0.134   9   11.6%  -5.3%
Maine                 1.921   -0.108   2    3.7% -24.2%
Wyoming               0.918    0.007   1   11.1%  82.8%
Hawaii                1.895   -0.040   2    7.5% -23.1%
New Mexico            2.887    0.037   3   11.4% -14.4%
Nevada                3.827    0.705   4   35.3%  -9.3% Could gain 5th in 2020
Pennsylvania         17.734   -1.213  18    2.9%  -1.9% Could lose 2 in 2020
Alabama               6.658   -0.220   7    6.4%  -3.0% Could lose 1 in 2020
West Virginia         2.604   -0.230   3    0.7%  -4.7% Could lose 1 in 2020
Nebraska              2.580   -0.106   3    5.4%  -3.8% Could lose 1 in 2020
Rhode Island          1.561   -0.131   2    0.5%  -4.9% Could lose 1 in 2020
New York             27.495   -1.777  28    3.2%  -0.6% Lucky to lose only 1 in 2010.
South Carolina        6.488    0.280   7   14.9%  -0.4% First change since 1930.
Montana               1.465   -0.014   1    8.8%   2.2% Just can't quite make it back.
Missouri              8.464   -0.182   9    7.6%  -0.2%
Arizona               9.460    1.532  10   31.2%  -0.2% AZ and WA are almost tied.
Washington            9.459    0.355  10   14.2%  -0.2%
Minnesota             7.448   -0.155   7    7.6%   0.2% Could still get lucky.
Oregon                5.441    0.140   5   12.8%   0.6% Could get 6th in 2020.
Delaware              1.350    0.043   1   14.1%  12.2% 2nd in 2050?
North Carolina       13.339    0.919  13   18.0%   0.7% Was lucky in 2000.  10th largest.
Florida              26.336    1.681  26   17.4%   0.1% Lost 27th (+2) in last year.
Louisiana             6.327   -0.584   6    0.6%   2.3%
Massachusetts         9.295   -0.510   9    4.2%   1.7%
Texas                35.274    3.110  35   20.5%   0.1% 36th if AZ, CA, FL drop.
New Jersey           12.265   -0.722  12    3.8%   1.4% Could lose 1 more in 2020.  11th
Iowa                  4.260   -0.281   4    3.0%   5.2%
Idaho                 2.257    0.200   2   21.2%  10.8% Perhaps could have 3rd in 2020
South Dakota          1.251   -0.016   1    8.3%  22.7%
California           52.243    0.001  53    9.9%  -0.1% Was lucky in 2000.
Ohio                 16.229   -1.288  16    1.8%   1.2% Lost 18th before 2001.
Oklahoma              5.226   -0.119   5    7.4%   4.8%
Mississippi           4.185   -0.230   4    4.1%   7.1%
Illinois             18.183   -0.979  18    4.3%   1.2% Will likely lose another in 2020
Virginia             11.171    0.241  11   12.3%   2.5% Outside chance of gain in 2020/
Colorado              7.159    0.505   7   18.3%   4.3% Could get 8th in 2020.
Arkansas              4.111   -0.042   4    8.8%   9.1%
Alaska                1.108    0.020   1   12.4%  42.3%
Kentucky              6.107   -0.147   6    7.3%   6.0%
Indiana               9.068   -0.323   9    6.1%   4.3%
Maryland              8.062   -0.122   8    8.2%   4.9%
North Dakota          1.037   -0.073   1    0.8%  54.9%
Utah                  4.010    0.529   4   26.9%  11.9% Sure 4th, 5th in 2020?
Georgia              14.009    1.373  14   21.9%   3.0% Another 1 or 2 in 2020 (8th)
Vermont               1.007   -0.057   1    2.3%  61.0%
Michigan             14.004   -1.332  14    0.3%   3.0% Drops to 9th largest.
Kansas                4.002   -0.175   4    5.2%  12.1%

[jimrtex]

There are five sets of divisors which can be used in the process of apportioning representatives among the States.  A quotient is calculated for each possible seat that a State may receive:

Q_{i n} = P_i / D_n

Where the quotient for State i for its n seat is equal to its population divided by the n divisor.

The sets of divisors are:

• D_n = n
• D_n = (n+n-1)/2    Arithmetic mean
• D_n = √n(n-1)       Geometric mean - currently used divisors
• D_n = 2n(n-1) / (2n-1)    Harmonic mean
• D_n = n-1

These are ordered from the method that is most favorable to large states, to the method that is most favorable to small states.

In practice, we do not calculate Q_{i 1} since all States are guaranteed at least 1 representative, and D₁ is 0 for the last 3 methods, which would result in division by zero.  Alternatively, we can define Q_{i 1} as

Q_{i 1} = P_USA + P_i

This ensures that

Q_{i 1} > Q_{i 2}

and more generally

Q_{i n} > Q_{i n+1} for n ≥ 1

That is, the quotients for each State are monotonically decreasing.

Q_{i 1} > Q_{j 2} for all i and j

That is every State is apportioned a first representative before any State is apportioned a second, and

If P_i > P_j, then Q_{i n} > Q_{j n}

That is the quotients for seat _n are ordered by population of the States.

We can divide the values of Q by a positive constant without having an effect on the ranking of the quotients, and in particular we can use the raw Q value of the quotient that is ranked 435_th.  We refer to this value q, as the quota.

Q'_{i n} = (P_i / q)  / D_n

A State will be apportioned n representatives if the following is true.

Q'_{i n+1} < 1 ≤ Q'_{i n}


The first method uses a divisor that is equal to n.

D_n = n

Then

Q_{i n} = P_i  / n

q = Q_{AZ 10} = 673,150

Q'_{i n} = (P_i / q)  / n

And

(P_i / q)  / (n+1) < 1 ≤ (P_i / q)  / n

(P_i / q) n / (n+1) < n ≤ P_i / q

Which is equivalent to saying that a State will be apportioned n representatives where n is the integer portion of P_i / q

q = 673,150, is the minimum congressional district size (other than for the single at-large district in ND, VT and WY).  673,150 is 95% of the ideal district size of P_USA / 435 = 709,495, and could be considered a minimum workload for a representative.  A State may not have another representative unless that representative could have a minimum workload.  In States like ME, NH, and RI, it would mean that the sole representative was pulling almost double shifts, while in CA representatives would have to work less than 10 minutes overtime before another representative would be apportioned to relieve their collective burden.

This method is most favorable to large States and would result in the apportionment of additional representatives to CA (+2), TX (+2), and NY, FL, IL, OH, and NC.  PA misses out because it is somewhat close to losing a 2nd seat under the current method, and this method would eliminate the risk, but not actually increase its apportionment.  10th most populous NC would gain a seat because it is near receiving a 14th representative under the current method, and this method would give it enough boost to finish above the line (429th).  States that would lose are RI, HI, ME, NH, NE, WV, SC, MO, and WA.  The four smallest would lose half their representation, while NE and WV would lose 1/3 of their representatives (both could well lose their 3rd representative under the current method in 2020).  Moderately large State WA and MO would drop below the line from 434th to 436th and 433rd to 439th, while SC would drop from 431st to 448th.  The 18th largest States would in general benefit from this method, but in some cases the benefit is tiny, and may not be true for specific population distributions (as it does not for WA and MO in 2010).  This method is the same a D'Hondt, excluding the guarantee of one representative for all States.


The second method uses a divisor that is equal to n-1.

D_n = n-1

Then

Q_{i n} = P_i  / n-1

q = Q_{LA 7} = 748,993

Q'_{i n} = (P_i / q)  / (n-1)

And

(P_i / q)  / n < 1 ≤ (P_i / q)  / (n-1)

(P_i / q) (n-1) / n < n-1 ≤ P_i / q

Which is equivalent to saying that a State will be apportioned n+1 representatives where n is the integer portion of
P_i / q

q = 748,993, is the maximum congressional district size.  748,993 is 105.5% of the ideal district size of P_USA / 435 = 709,495, and could be considered a maximum workload for a representative.  If there is even a minute of work to be performed beyond 8 hours, an additional representative would be apportioned.  The total workload would then distributed among all representatives.  In SD this would mean that the 2 representatives could take off the rest of the day after lunch.  Larger States would lose representatives as their representatives would be expected to work up to a 1/2 hour of uncompensated overtime as they perform more than 1/435 of the congressional workload.

This method is most favorable to small States and would result in the apportionment of additional representatives to SD, DE, MT, ID, IA, OR, LA, and MN.  IA, LA, and MN would retain the seat that they are projected to lose under the current method, while the others would gain a seat.  Losers would be CA (-3), TX, NY, PA, AZ, and WA.  This would eliminate anticipated gains for TX, AZ, and WA, and would be true decreases for CA, NY, and PA.

In general, 35 States would benefit from this method, ensuring easy passage in the Senate including cloture votes, but might be defeated on a 286:149 vote in the House if all representatives voted in their self interest.


The third method uses a divisor that is the arithmetic mean of n and n-1

D_n = (n+n-1)/2  = (2n-1)/2 = n - 1/2

Then

Q_{i n} = P_i  / (n - 1/2)

q = Q_{FL 27} = 708,009

Q'_{i n} = (P_i / q)  / (n-1/2)

And

(P_i / q)  / (n+1/2) < 1 ≤ (P_i / q)  / (n-1/2)

(P_i / q) (n-1/2) / (n+1/2) < n-1/2 ≤ P_i / q

(P_i / q) (2n-1)/(2n+1) + 1/2 < n ≤ P_i / q + 1/2

Which is equivalent to saying that a State will be apportioned n+1 representatives, if the
integer portion of P_i / q is n, and the fraction is greater than or equal to 1/2.

q = 708,009 which is 99.7% of the ideal district size of P_USA / 435 = 709,495, and could be considered an average district size.  If we did not insist on apportioning exactly 435 representatives, we could simply divide each State's population by the ideal, and give an additional representative to each State whose fraction was greater than 1/2.  For example a State whose share of the national population was between 12.5 / 435 and 13.5 / 435, would be apportioned 13 representatives, regardless of the population of the other States, except as to their effect on the total population.

In 2010, such a method would result in a House of 429, as SC, CA, AZ, WA, MO, and FL would lose a representative.  This is a matter of the coincidence of an inordinate number of States having a fraction slightly less than 0.500.  In other decades more than 435 representatives would have been apportioned.  In 2000 there would have been 433 representatives.  NC and UT were fighting for a seat that neither deserved on the basis of their share of the total population, and no one noticed that CA had sneaked off with the 434th, so used were they to a constantly growing delegation.

There would be only a small difference between using the arithmetic mean and geometric mean, with RI losing its 2nd seat (its population share is 1.485 / 435) and FL gaining another seat.  FL would gain the seat by advancing from 436th (geometric mean) to 435th (arithmetic), which is entirely due to the drop by RI.

RI would argue the plan was unfair, since its two districts would only be about 25% below the national average, while a single at large district would be almost 50% larger.  On the other hand, if instead of measuring district size (or persons/representative), one measured representation or responsiveness (or representatives/person) then voters in RI would go from having a representative who was 33% more likely to be responsive than the average representative to one who was 33% less likely to be responsive.

Only relatively small States (those with less population than Oregon) would not benefit from this method.  While the divisor for a 6th seat would increase from 5.477 to 5.500, the quotients to qualify for the 435th seat would be smaller.   In a self-interested vote, the Senate might approve a change 53:47, but it would be overwhelmingly approved by the House 377:58.  Were it first approved in the House with a lot of noise about compensating for over-representation in the Senate, it could provoke more opposition in the Senate.

This method is the same as St.Lague except with the guarantee of a seat for every State.  In 2010 there would be no difference since WY has a population that is 77% of the ideal district size, far beyond the 50% needed.  In St.Lague the divisors are conventionally expressed as 2n - 1, rather than (2n-1)/2.  This makes hand calculation slightly simpler, but makes no difference in the result.

[jimrtex]

The fourth method uses a divisor that is the harmonic mean of n and n-1

D_n = 2n(n-1) / (2n-1) = n -  n / (2n-1)

Then

Q_{i n} = P_i  / (n - n / (2n-1))

q = Q_{WA 10} = 710,481

Q'_{i n} = (P_i / q)  / (n - n / (2n-1))

And

(P_i / q)  / (n+1- (n+1)/(2n+1) < 1 ≤ (P_i / q)  / (n - n / (2n-1))

(P_i / q) (n+1- (n+1)/(2n+1)) / (n+1- (n+1)/(2n+1) < n - n / (2n-1) ≤ P_i / q

Which is equivalent to saying that a State will be apportioned n+1 representatives, if the
integer portion of P_i / q is n, and the fraction is greater than or equal to n/(2n+1), or 1/3, 2/5, 3/7, ....  The fraction converges on 1/2.

q = 710,481 which is 100.1% of the ideal district size of P_USA / 435 = 709,495, and could be considered an average district size.  If we did not insist on apportioning exactly 435 representatives, we could simply divide each State's population by the ideal, and give an additional representative to each State whose fraction was greater than n/(2n-1).  For example a State whose share of the national population was between (12+12/25) / 435 and (13+13/27) /435, would be apportioned 13 representatives, regardless of the population of the other States, except as to their effect on the total population.  Since the fractional thresholds are slightly less than 1/2, this tends to produce a slightly larger House (438.463 in 2010).   If we increase the ideal quota upward slightly (to 715,228), we would tend to have 435 members based on the overall population distribution.

In 2010, such a method would result in a House of 430, as SC, NY, MO, AZ, and WA would lose a seat, much the same as if a threshold of 1/2 were used.

There would be only a small difference between using the harmonic mean and the geometric mean, with MT gaining its 2nd seat (its population share is 1.383 / 435), and CA losing a seat.  CA would gain the seat by dropping from 435th (geometric mean) to 437th (arithmetic), which is due to MT advancing 17 places and MN squeezing into 435th place.

MT would argue this plan was fairest, since the deviation of district sizes from the ideal would be minimal.  A State whose single at-large representative represented slightly more than 33% of the ideal, would switch to having 2 districts that were slight less than 33% below the ideal.  In 1990, MT lost that argument before the US Supreme Court, which ruled that the apportionment method used is at the discretion of Congress.

Only relatively small States (those with less population than Oregon) would benefit from this method.  While the divisor for a 6th seat would decrease from 5.477 to 5.455, the quotients to qualify for the 435th seat would be larger.   In a self-interested vote, the Senate might defeat a change 48:52, but it would be overwhelmingly turned down by the House 65:370,


The fifth method is that is currently used, where the divisor that is the geometric mean of n and n-1

D_n = √n(n-1)

Then

Q_{i n} = P_i  / √n(n-1)

q = Q_{CA 53} = 709,065

Q'_{i n} = (P_i / q)  / √n(n-1)

And

(P_i / q)  / √n(n+1) < 1 ≤ (P_i / q)  / √n(n-1)

(P_i / q)  / √(n+1)(n-1) < √n(n-1) ≤ (P_i / q)

Which is equivalent to saying that a State will be apportioned n+1 representatives, if the
integer portion of P_i / q is n, and the value of P_i / q ≥ √n(n+1).  The threshold values are √(1)(2), √(2)(3), √(3)(4),  ..., or √2, √6, √12, ..., or 1.414, 2.449, 3.464, ..., which converges on n + 1/2.

q = 709,065 which is 99.9% of the ideal district size of P_USA / 435 = 709,495, and could be considered an average district size.  If we did not insist on apportioning exactly 435 representatives, we could simply divide each State's population by the ideal, and give an additional representative to each State whose quotient exceeded √n(n+1).  For example a State whose share of the national population was between √156 / 435 and √182 /435 (√146 = 12.490 and √182 = 13.491), would be apportioned 13 representatives, regardless of the population of the other States, except as to their effect on the total population.  Since the fractional thresholds are slightly less than 1/2, this tends to produce a slightly larger House (436.857 in 2010).   If we increase the ideal quota upward slightly (to 712,550), we would tend to have 435 members based on the overall population distribution.

In 2010, such a method would result in a House of 429, as NY, SC, AZ, MO, WA, and CA would lose a seat, much the same as if a threshold of 1/2 were used.

When the geometric mean is used, in the State receiving the 435th seat, the following relationship will hold:

dist_n-1 / q = q / dist_n

For 2010, CA receives the 435th seat, and q = 709,065, dist₅₂ = 715,851 and dist₅₃ = 702,344.   715,851 / 709,065 = 1.0096 = 709,065 / 702,344. 

[Antonio the Sixth]

So Oregon is not gaining a seat anymore and instead Washington will ? Weird.

And glad to see Texas will gain only 3 !

[muon2]

So Oregon is not gaining a seat anymore and instead Washington will ? Weird.

And glad to see Texas will gain only 3 !

WA and OR are both close to the line, but the projections right now favor WA. Similarly, TX and AZ may be fighting for the last seat. I have it going to AZ right now, but many other projections give it to TX. That would give them +4.

[Antonio the Sixth]

So Oregon is not gaining a seat anymore and instead Washington will ? Weird.

And glad to see Texas will gain only 3 !

WA and OR are both close to the line, but the projections right now favor WA. Similarly, TX and AZ may be fighting for the last seat. I have it going to AZ right now, but many other projections give it to TX. That would give them +4.

Well, interesting trends anyways. It has certainly been said before, but it's still interesting to notice how States gaining  seats are traditionally Republican States but with a democratic trends in the last decades... I wonder which sort of elections it will give us in the future.

And I also wonder when will demographic erosion stop in the Northeast and Midwest States. It seems to be diminishing, compared to 1980 or 2000.