Census population estimates 2011-2019

[Minnesota Mike]

I've been watching the census estimates throughout the decade and I've seen California move around a bit. To those that pay attention more closely, what are the odds or what will it take for the creation of CA-54?

Losing a seat is more likely than gaining one. 

 

[Minnesota Mike]

So it's quite likely both MT and RI have just 1 CD?

RI will almost certainly lose a seat but MT will be right on the threshold of gaining a seat. Right now the projection using short term trends has them gaining a seat, the projection using long term trends has them staying at one seat (FWIW I prefer using short term trends).

BTW Congress uses a different method of apportioning seats ( the Huntington–Hill method) than jimrtex does (The Vinton or Hamilton method) that can produce slightly different results.

https://en.wikipedia.org/wiki/United_States_congressional_apportionment

[Minnesota Mike]

So it's quite likely both MT and RI have just 1 CD?

RI will almost certainly lose a seat but MT will be right on the threshold of gaining a seat. Right now the projection using short term trends has them gaining a seat, the projection using long term trends has them staying at one seat (FWIW I prefer using short term trends).

BTW Congress uses a different method of apportioning seats ( the Huntington–Hill method) than jimrtex does (The Vinton or Hamilton method) that can produce slightly different results.

https://en.wikipedia.org/wiki/United_States_congressional_apportionment

I use Huntington-Hill. My use of "raw" may have misled you. I used a slightly cooked version.

The divisor methods don't nominally use a quota. We just divide state populations by the successive divisors and rank the quotients.

The divisors for Huntington-Hill are:

sqrt(n*(n+1) = 1.414, 2.449, 3.464, ... 52.498, ...

In 2010, California had a population 37,254,518 (disregarding the overseas population).

For California's 53 seat, we divide the state population by sqrt(52*53) = 52.498

37254518 / 52.498 = 709642.051

If 709642.051 is among the 435 largest quotients then California gets a 53rd seat which it did. But we notice that the quotients near the 435th seat are about equal to the quota for the USA population divided by 435 = 708,408. But for reasons, I'll explain later, I'll divide by 711,428 (Q' adjusted quota).

We divide every state's population by 711,428.

q' = P / Q'

P'(California) = 52.366

We can then divide by the divisors, so 52.366 / 52.498 = 0.997. Values close to one represent states just below or above the 435th seat. We still have to rank these values, to determine which are rounded up or not. If we were doing independent rounding, we could simply look at the value, and since it is less than 1, round down. But if we did independent rounding, we could not guarantee 435 representatives.

Dividing by a positive constant does not change the relationship for ranking purposes:

if a > b and c > 0 then a/c > b/c

For Webster's method, a state with population p is entitled to r representatives, where
Q = P_USA/N, where P_USA is the total population of the 50 states, and N is the number of representatives (435 in this case).

p/Q = r

We can look at the value of R, and say that a state should have r representatives, but that is only true if we are apportioning fractional representatives. But it nonetheless correct for the raw share of the population. If a state has 27.235 / 435 of the population, it should have 27.235 representatives, except for the silly notion that representatives must be whole persons.

For Huntington-Hill

p/Q = sqrt((r-1/2)*(r+1/2))

To simplify our expression, we will use q = p/Q where q is a quotient.

q = sqrt(r-1/2)*(r+1/2))

Squaring both sides and multiplying the two terms under the radical.

q² = r² - 1/4

Solving for r,

r = sqrt(q² +1/4)

This is the raw entitlement under Huntington-Hill. For example, a state with sqrt(2)/435 of the total population is entitled to 1.5 representatives.  1.5 = sqrt(2 + 1/4)

But if we do this for all states, and sum them up, we will end up with 436.841 representatives. But we can adjust our quota, so that it is Q' = P_USA/(435*(435/436.841). This will yield 435.015 representatives. Adjusting to 435/436.856 produces the expected 435 representatives.

So the "raw" numbers

r = sqrt(q'² +1/4)

Where q' = p/Q'

And Q' is the adjusted quota.

For 2010, it is 711,428. The adjusted quota is dependent on the overall distribution of populations, but it can be calculated easily and converges quite quickly. The adjustment is increasing slightly

436.856/435 for 2010
436.873/435 for 2017 estimate.
436.881/435 for 2020 projected.

This increase indicates a greater small state bias. Not unexpected considering that the 2nd and 3rd most populous states are fast gainers, and the most populous state is an average gainer.

Among states with 3 or fewer representatives:

AK, SD, WY, VT, HI, RI, NH, ME, NM, NE, and WV are losing population share.
MT is staying even
DE, ID, ND* are gaining population share, but ND is quite iffy, and DE is moderate.

While the difference is 0.43%, the cost to California is 0.43% * 52.5 = 0.23 representatives.

Anyhow, the raw numbers provide an estimate of how many representatives a state should have if independent rounding were done and the distribution is based on Huntington-Hill. It is particularly useful for seeing temporal trends (e.g New Jersey is losing about 1/2 a representative per decade).

I do use it as an estimate of the number of seats a state should have. If a state is entitled to n.xxx representatives (as represented by a mixed decimal fraction), then it should have either n or n+1 (with a minimum of one). Summing the values of n, I can calculate the number of guaranteed seats without rounding. For 2020 projections this is 412 seats, leaving 23 seats (435-412) to be apportioned by rounding. I then divide the projected population of each state by the divisor for the next seat; sqrt(n*(n+1)) and assign the 23 final seats based on the largest quotients. This does not determine the 413th-435th seats assigned, but rather that a state will be in the top 435.

Note that the divisor methods do not not guarantee an apportionment of between n and n+1 seats. It may be outside that range, for example n-1 or n+2. This is known as a quota violation, and is a known flaw of Huntington-Hill and other divisor methods. This requires an unusual (rare) distribution of fractions, and modeling for the US indicates that it will be quite rare for the USA, because of the large number (50) of entities getting seats.

It may be quite common when there are only a few entities, and one is relatively quite large. An example is the British House of Commons, where there are only four apportionment entities (England, Scotland, Wales, and Northern Ireland), and England has 80%+ of the population.

Got it.