USA 2020 Census Results Thread (Release: Today, 26 April)

[prag_prog]

https://twitter.com/kkondik/status/1386842867394617347

[cinyc]

One note: I keep on saying PR had the highest error rate. This was based on a Twitter Demographer forward allocating from the 7/1/19 estimates. For some reason, PR wasn’t in the 7/1/20 estimates, according to the demographer. In a publication, Census actually put out a new 4/1/20 estimate. They largely match mine, but not in PR, where the error dropped into the +3s.

https://www.census.gov/content/dam/Census/library/working-papers/2021/demo/pop-twps0104.pdf

[jimrtex]

I am almost certain that Webster's method would flip the seat from Minnesota to New York.

Actually, under Webster's method (or rather Major Fractions as used in 1910 and 1930, having a fixed House size rather than having a fixed ratio as was done in the beginning of our country's history (with a largest remainder method with a bunch of paradoxes in between fixed ratio and fixed House size divisor methods)), Minnesota would still have 8 seats but seats would have "moved" from Montana and Rhode Island to New York and Ohio.  Minnesota would have lost a seat under the Harmonic Mean ("Dean") divisor method to Idaho (which would have gained a 3rd seat), interestingly enough.

There is probably a value of p in the generalized or power mean between 0 (geometric mean; equal proportions) and 1 (arithmetic mean; major fractions) that would just flip a seat from Minnesota to New York, but I haven't tested that yet.  Montana got the second-to-last seat IRL so I can't be sure it wouldn't lose one before Minnesota without checking.

1 / 2.000006951 would tie the two.

The quotients using the square root for Minnesota and New York are only 4 apart while Montana is 4501 greater than Minnesota.

Using 1 / 2.000006951 would reduce the gap between Montana and Minnesota-New York to 4497.

Ranking methods and a fixed house size are fundamentally flawed. If Minnesota had lost the seat it would have increased its share of the USA population while simultaneously reducing it share of the USA representation.

[cinyc]

One note: I keep on saying PR had the highest error rate. This was based on a Twitter Demographer forward allocating from the 7/1/19 estimates. For some reason, PR wasn’t in the 7/1/20 estimates, according to the demographer. In a publication, Census actually put out a new 4/1/20 estimate. They largely match mine, but not in PR, where the error dropped into the +3s.

https://www.census.gov/content/dam/Census/library/working-papers/2021/demo/pop-twps0104.pdf

Yeah, that series (the 2020 population estimates which don't take into account the 2020 census) for Puerto Rico will be released on May 04, along with all the county populations (same series)

I still think Puerto Rico will have a high error rate. After all, New Jersey and New York saw the two highest error rates in the country (as seen below), and they're two states known for large Puerto Rican populations.

It did. Under the 4/1/20 estimates in Census' chart, PR was underestimated by 3.7%. See Table 3 on Page 6 at the link.

[cinyc]

NYer overcount due to COVID?

https://twitter.com/lymanstoneky/status/1387161150522073088

[RFK Jr.’s Brain Worm]

The house size needs to be expanded. The undercount is so obvious here

If just nineteen seats were added to the house, no states would lose any seats this redistricting cycle (I believe WV-3 is the 454th seat). You’d have thought that some representative from a state that’s losing out would want to put forward a bill to expand the house a bit so they can keep their seat...

That was common practice in the 19th century. I did a thread about what would have happened had the practice been codified and 435 was never fixed as the number of seats. I've just posted an update for that thread with the effect of the 2020 Census.

With the release of the 2020 apportionment populations I can continue the series. In this timeline the 1929 act never happened to lock in 435 members in the House, but instead codified what had been common practice since the Civil War. Initially no state could lose a seat due to reapportionment, but that was modified by the 1947 Act such that no state that gains population over the decade can lose seats. In 2010 the total in the House was 1140 members.

In the 2020 Census IL, MS and WV lost population and both IL and WV lose a seat as OH is the last to be brought up to their status quo. VT and WY remain with 2 seats, so DC continues to get 4 electors. There are now 1193 members in the House and the average district has 278 K inhabitants.

On the flipside of this, do you want 1200 members of the House? The House has become so incredibly centralized in power in each party that there's only about 30 to 40 people in the House total where their opinion matters (and with the removal of power from Committee chairs the past few decades in favor of party leadership, you can argue it's even less), so from a purely legislating point of view, increasing the size of the House from 435 to 1193, you're adding 758 backbenchers that are only going to vote as their leadership tells them to.

OTOH a smaller constituency makes it easier to vote the district and not the party. That's especially true if anti-gerrymandering laws keep district lines from carving up identifiable communities and piecing together distant fragments. That would make it easier for a representative to be known as an individual and not merely as a label.

How many elected representatives are there in state legislatures elected on third-party labels? There's 1 in Wyoming that is a Libertarian.

Vermont has (iirc) 7 Progressive Assemblymembers and 2 Progressive Senators. There’s some independents interspersed throughout the country (Alaska has a few, I think; California has one in the Assembly).

On a federal level (based on the quoted map), I could see a few in some of the bigger cities, possibly a Progressive in Vermont, and probably a few independents. You might see a general rise in third parties, if nothing else than in vote share, but I would be shocked if there weren’t at least a few third party members or independents.

Yep that's how regional parties and third parties are able to hold power in countries like England and Germany who use FPTP when you have smaller districts third parties have more opportunity to find favorable ground and run strong campaigns. This is why we see third parties elected at the state leg level a very small amount but still at the federal level we only get third-party reps by switches. Id wants to make the senate a mostly ceremonial house and expand the house to 700-1000 seat houses and use RCV for presidential elections. 

Germany uses MMP (like NZ and a few others), but yeah. Smaller districts absolutely help smaller parties. Unlike the UK and Canada, we don’t have regionalist parties.

Canada is something of an oddity with how well the NDP does (they were the opposition from 2011-2015) in FPTP, though I’d say that some of that has to do with smaller riding sizes than our districts here. Another thing that might help the NDP is that they’re the only federal party with matching provincial parties (outside of Quebec). I’d imagine that the organization helps there.

The Liberal Democrats are a rough equivalent in terms of being the (now historical, tbh) half a party in the 2.5 party systems of Canada and the UK, but they’ve been on the decline for the most part since 2010.

[jimrtex]

Census put out at least 2 blog posts about the quality of the count that we might find interesting:

https://www.census.gov/newsroom/blogs/random-samplings/2021/04/examining-operational-metrics.html

https://www.census.gov/newsroom/blogs/random-samplings/2021/04/comparisons-to-benchmarks-as-a-measure-of-quality.html

Some takeaways for the tl;dr crowd:

-65.28% of census addresses were resolved by self-response, vs 61.05% in 2010.
-Only 55.48% of non-response follow up households “were enumerated with a household member; 26.07% were resolved with a proxy respondent, such as a neighbor, building manager or landlord; and 18.44% were enumerated using high-quality administrative records. The use of administrative records was new to the 2020 Census, but the proxy rate is comparable to the 2010 Census, in which 24.71% of occupied households in Nonresponse Followup were enumerated by a neighbor or other knowledgeable person. The proportion enumerated by a household member was 74.88% in 2010.”

In other words, 19% fewer NRFU responses were from a knowledgeable person. Could this have had an effect on the unduplication efforts? Maybe. Census reported a few days ago that there were more duplicates this year than in 2010, and they took additional steps to try to resolve them:

https://www.census.gov/newsroom/blogs/random-samplings/2021/04/how_we_unduplicated.html

Any other thoughts from those more knowledgeable than me?

Nothing really stands out on these metrics, you can compare state to state or state to USA etc. Perhaps if you got them in a spreadsheet you could try to identify outliers.

https://www.census.gov/library/visualizations/interactive/2020-census-operational-quality-metrics.html

Something I had not realized that most of the proxy responses were either that the housing unit was vacant or deleted (did not exist). The Census Bureau in 2020 identified bunches of hypothetical housing units (around 150 million total, even though the ACS number is closer to 130 million). If you are looking for a sample, you can skip locations that are only 2% likely to have anyone living in them. If you want a complete count you check whether they exist.

Lots of "vacant" housing units are vacation/summer homes. Persons per housing unit in Hamilton County NY is under 1 per HU. There may be non-rentable apartments in New York, and even more in other cities such as Detroit.

Unresolved housing units were highest in several distinct areas:

New Mexico
Black Belt: Louisiana to Georgia
West Virginia
Northeast, including New York, Massachusetts, and Vermont

There must be economic reasons for the first three. I don't know what it is for the Northeast. Perhaps there are more dual use areas (rooms behind or above stores) that are less likely to occur in the West and South where residential and commercial areas are more distinct.

New York and other northeast states had distinctly low self-response rates during the initial census period, undoubtedly related to Covid19. Areas in the West and South had lower rates, but that is because they had not received census forms in the mail, and Update/Leave was delayed.

You could do Self Response without receiving a form, but it was somewhat harder to do.

Later, during Update/Leave and NRFU, New York did catch up some in self response. One could self respond even if it was because an enumerator had pushed you to do it. "I mailed it last week. It will probably arrive in the next week. Or two." (after enumerator leaves)
"Do you remember where we put the census forms?" "It is in the pizza box with the insurance policies, car registration, and lottery tickets. If it is not there, it is probably in the clothes hamper."

Outreach might be more successful in NYC. There will be persons who can not be reached in any other way. Someone from Chad might only respond if someone from the New York Chadean Culture Center (or whatever it is known as in French) helps them to respond. It is possible that these assistant push to report persons who would otherwise not be reported.  It might be the cousin who was visiting, and the assistant assures the respondent that the Census Bureau will sort it out, and that it won't be reported to ICE.

New Yorkers may be less willing to admit too much knowledge about a neighbor. Some of these housing units may be imputed as occupied, when they were actually vacant.

[cinyc]

Here's census' Operational data state spreadsheet:

https://www2.census.gov/programs-surveys/decennial/2020/data/operational-quality-metrics/census-operational-quality-metrics-release_1.xlsx

FAQs:
https://www2.census.gov/programs-surveys/decennial/2020/data/operational-quality-metrics/operational-quality-metrics-faqs_release-1.pdf

And technical documentation:
https://www2.census.gov/programs-surveys/decennial/2020/data/operational-quality-metrics/operational-quality-metrics-technical-documentation_release-1.pdf

I'm still not entirely sure what all this means, but New York was number 1 in the Unresolved, person unduplication category, which I think means they removed the most duplicates from NYS. NYS was about average in the makeup of NFUR method - 55% interview/27% Proxy/19% Administrative Record. It has the 10th highest percentage of households considered occupied using administrative records, but was 45th in administrative record vacancies.

Edit: now, I realize this is just the underlying data behind the data viz you linked.

[ꙮ]

I'm not sure how large of an effect this is, but I think a NY overcount could have something to do with colleges. According to the Census Bureau, college students living on campus should have been counted at their dorms by their colleges, and not included at their family's address. I was included by my family in Utica because I was at home on April 1, which means I was counted twice. I talked to a couple other students I know and they said the same thing.

As of 2018, there were 732,000 full time undergraduate college students studying in New York. That's a lot of potential error.

[cinyc]

I'm not sure how large of an effect this is, but I think a NY overcount could have something to do with colleges. According to the Census Bureau, college students living on campus should have been counted at their dorms by their colleges, and not included at their family's address. I was included by my family in Utica because I was at home on April 1, which means I was counted twice. I talked to a couple other students I know and they said the same thing.

As of 2018, there were 732,000 full time undergraduate college students studying in New York. That's a lot of potential error.

Census was supposedly working with the colleges to reassign students to their colleges. Whether they successfully did so will really only be known when we see the redistricting file.

[○∙◄☻¥tπ[╪AV┼cVê└]

In the media focus on NY ending up 89 people short, they miss the fact that had MN counted 26 fewer people they would have lost the seat to NY. The asymmetry of the differences is due to the size of sqrt(26*27) in NY compared to sqrt(7* in MN.

Extra credit algebra problem:

How many persons would have to move from MN to NY to flip the seat.

I think it's 20.

[patzer]

Have checked with a spreadsheet and you are correct.

[cinyc]

Have checked with a spreadsheet and you are correct.

Who is correct?

[patzer]

Have checked with a spreadsheet and you are correct.

Who is correct?

Poster above me (○∙◄☻¥tπ[╪AV┼cVê└). 20 people moving from MN to NY would be needed. I intended to quote the comment but forgot.

[Kevinstat]

I am almost certain that Webster's method would flip the seat from Minnesota to New York.

Actually, under Webster's method (or rather Major Fractions as used in 1910 and 1930, having a fixed House size rather than having a fixed ratio as was done in the beginning of our country's history (with a largest remainder method with a bunch of paradoxes in between fixed ratio and fixed House size divisor methods)), Minnesota would still have 8 seats but seats would have "moved" from Montana and Rhode Island to New York and Ohio.  Minnesota would have lost a seat under the Harmonic Mean ("Dean") divisor method to Idaho (which would have gained a 3rd seat), interestingly enough.

There is probably a value of p in the generalized or power mean between 0 (geometric mean; equal proportions) and 1 (arithmetic mean; major fractions) that would just flip a seat from Minnesota to New York, but I haven't tested that yet.  Montana got the second-to-last seat IRL so I can't be sure it wouldn't lose one before Minnesota without checking.

1 / 2.000006951 would tie the two.

The quotients using the square root for Minnesota and New York are only 4 apart while Montana is 4501 greater than Minnesota.

Using 1 / 2.000006951 would reduce the gap between Montana and Minnesota-New York to 4497.

I didn't understand quite what you were saying here.  Some slight weight to the geometric mean, perhaps?  Feel free to answer "offline" if you wish (it might be of limited interest, even among those who frequent apportionment discussions on this forum).

The value of p in the power mean of (# of seats a state already has been "awarded") and (# of seats a state is "going for") where Minnesota and New York would be tied for the last seat is p ≈ 0.0021.  Minnesota and Montana would be tied for the last seat at p ≈ 0.1017.  Minnesota oscillates between 7 and 8 seats several times as p increases from −∞ to ∞ (or decreases from ∞ to −∞).

Ranking methods and a fixed house size are fundamentally flawed. If Minnesota had lost the seat it would have increased its share of the USA population while simultaneously reducing it share of the USA representation.

All in the eye of the beholder.  Scholarly works will sometimes talk of divisor methods with a fixed house size (or at least fixed as far as that being the "key number" and not the ratio) as being the gold standard.  It doesn't have some of the paradoxes that largest remainder methods have.

[StateBoiler]

The house size needs to be expanded. The undercount is so obvious here

If just nineteen seats were added to the house, no states would lose any seats this redistricting cycle (I believe WV-3 is the 454th seat). You’d have thought that some representative from a state that’s losing out would want to put forward a bill to expand the house a bit so they can keep their seat...

That was common practice in the 19th century. I did a thread about what would have happened had the practice been codified and 435 was never fixed as the number of seats. I've just posted an update for that thread with the effect of the 2020 Census.

With the release of the 2020 apportionment populations I can continue the series. In this timeline the 1929 act never happened to lock in 435 members in the House, but instead codified what had been common practice since the Civil War. Initially no state could lose a seat due to reapportionment, but that was modified by the 1947 Act such that no state that gains population over the decade can lose seats. In 2010 the total in the House was 1140 members.

In the 2020 Census IL, MS and WV lost population and both IL and WV lose a seat as OH is the last to be brought up to their status quo. VT and WY remain with 2 seats, so DC continues to get 4 electors. There are now 1193 members in the House and the average district has 278 K inhabitants.

On the flipside of this, do you want 1200 members of the House? The House has become so incredibly centralized in power in each party that there's only about 30 to 40 people in the House total where their opinion matters (and with the removal of power from Committee chairs the past few decades in favor of party leadership, you can argue it's even less), so from a purely legislating point of view, increasing the size of the House from 435 to 1193, you're adding 758 backbenchers that are only going to vote as their leadership tells them to.

OTOH a smaller constituency makes it easier to vote the district and not the party. That's especially true if anti-gerrymandering laws keep district lines from carving up identifiable communities and piecing together distant fragments. That would make it easier for a representative to be known as an individual and not merely as a label.

How many elected representatives are there in state legislatures elected on third-party labels? There's 1 in Wyoming that is a Libertarian.

There are many in state legislatures who act as independents or even as if they were in a different party. There was a recent state rep in IL who voted more like a conservative libertarian than the R label he ran under. Part of the problem is ballot access for anyone other than an R or D in IL. Ballot access for independents and third parties is easier here at local races, and it's not unusual to see an independent slate beat a party in township races.

I'm a county party chair of a third party so I'm more gung-ho about it than you are by definition, but this is a false argument. It does not matter in the argument you are making how they are voting, it matters what the people elect them as. Because if you're for example a libertarian Republican, you're still a Republican at the ballot box. Not all people are going to automatically switch your vote to continue backing you if you changed your party label or became independent because education about who is on the ballot and what they stand for considerably declines really once you get past any statewide race in this country and becomes reflexively about voting for a party ID.

[Tintrlvr]

I'm not sure how large of an effect this is, but I think a NY overcount could have something to do with colleges. According to the Census Bureau, college students living on campus should have been counted at their dorms by their colleges, and not included at their family's address. I was included by my family in Utica because I was at home on April 1, which means I was counted twice. I talked to a couple other students I know and they said the same thing.

As of 2018, there were 732,000 full time undergraduate college students studying in New York. That's a lot of potential error.

This happens every Census, though. I was double-counted in 2010 because I filled out a Census form in Massachusetts, where I was in college, but I later found out my parents also listed me as living at home in New Jersey on their Census form, thinking that was the right place for me to be recorded.

This Census, my double-count would have disappeared, vaporizing one Northeasterner.

[muon2]

I'm a county party chair of a third party so I'm more gung-ho about it than you are by definition, but this is a false argument. It does not matter in the argument you are making how they are voting, it matters what the people elect them as. Because if you're for example a libertarian Republican, you're still a Republican at the ballot box. Not all people are going to automatically switch your vote to continue backing you if you changed your party label or became independent because education about who is on the ballot and what they stand for considerably declines really once you get past any statewide race in this country and becomes reflexively about voting for a party ID.

I generally agree with that, but there are three townships near me that have been successfully and repeatedly won by third-party slates. I've worked closely with them at times, and advised them on how to navigate some of the ballot access, vacancy and candidate replacement laws.

I've learned a bit about how they have made it work. Most townships have partisan elections. Each of these three is based on a local party that only exists in the one township, and the slates are actually a mix of D, R, and I individuals based on my personal acquaintance with them. Their success has been based on the smaller size of their constituencies - 20 K to 50 K, and once in office they've been aided by incumbency. Educating voters is much easier on a small scale, especially if it seems to the voters that it is a local, grassroots effort.

[gerritcole]

we need 1001 house members and 150 senators asap

[jimrtex]

Here's census' Operational data state spreadsheet:

https://www2.census.gov/programs-surveys/decennial/2020/data/operational-quality-metrics/census-operational-quality-metrics-release_1.xlsx

FAQs:
https://www2.census.gov/programs-surveys/decennial/2020/data/operational-quality-metrics/operational-quality-metrics-faqs_release-1.pdf

And technical documentation:
https://www2.census.gov/programs-surveys/decennial/2020/data/operational-quality-metrics/operational-quality-metrics-technical-documentation_release-1.pdf

I'm still not entirely sure what all this means, but New York was number 1 in the Unresolved, person unduplication category, which I think means they removed the most duplicates from NYS. NYS was about average in the makeup of NFUR method - 55% interview/27% Proxy/19% Administrative Record. It has the 10th highest percentage of households considered occupied using administrative records, but was 45th in administrative record vacancies.

Edit: now, I realize this is just the underlying data behind the data viz you linked.

The app with the statistics was slow and kept resetting so I appreciate the spreadsheets. I was going to note that Louisiana had the highest non-resolution, but saw that you had specifically noted unresolved due to unduplication for New York.

In Louisiana the unresolved addresses were due to the inability to determine which pile of lumber was the former house, or even get into the hurricane areas.

New York was also a bit high in the Other Occupied. "Other" includes Rural Alaska and Update Enumerate, so Alaska was high in those categories. Rural Alaska was when enumerators started going out to the bush in January, when they hoped to catch people before they started moving around after the sun comes up. Update Enumerate was only used in Alaska and extreme Northern Maine. Unlike Update Leave, where the enumerator leaves the census form, in Update Enumerator does the enumeration on the first visit.

So for the other 48 States, "Other" incorporates "SRQA" and "Cov Imp". I assume SRQA is Self-Response Quality Assurance, and "Cov Imp" is coverage imputation. I don't know what this implies for New York.

If they receive a self response from an address, it is going to be assumed that the address exists and somebody lives there. It would be taken off the NRFU workload. There are conditions that could kick it back out, but the whole idea is that these don't need to be scrubbed down, and they can concentrate there efforts on the 30% to 40% or more which they don't have.

It is interesting that they got some vacant/delete from self response. I suppose this would be from people who control the mailbox, who responded that nobody lived there, or in some rare cases that the building no longer exists.

If they get another response for the same people from a different address, they first half to determine/guess which is the correct one. Since this was after field operations they might not be able to check back (or maybe they can?). So does unresolved mean that they are pretty sure, but not 100% sure that the other address was vacant on April 1.

[jimrtex]

I am almost certain that Webster's method would flip the seat from Minnesota to New York.

Actually, under Webster's method (or rather Major Fractions as used in 1910 and 1930, having a fixed House size rather than having a fixed ratio as was done in the beginning of our country's history (with a largest remainder method with a bunch of paradoxes in between fixed ratio and fixed House size divisor methods)), Minnesota would still have 8 seats but seats would have "moved" from Montana and Rhode Island to New York and Ohio.  Minnesota would have lost a seat under the Harmonic Mean ("Dean") divisor method to Idaho (which would have gained a 3rd seat), interestingly enough.

There is probably a value of p in the generalized or power mean between 0 (geometric mean; equal proportions) and 1 (arithmetic mean; major fractions) that would just flip a seat from Minnesota to New York, but I haven't tested that yet.  Montana got the second-to-last seat IRL so I can't be sure it wouldn't lose one before Minnesota without checking.

1 / 2.000006951 would tie the two.

The quotients using the square root for Minnesota and New York are only 4 apart while Montana is 4501 greater than Minnesota.

Using 1 / 2.000006951 would reduce the gap between Montana and Minnesota-New York to 4497.

I didn't understand quite what you were saying here.  Some slight weight to the geometric mean, perhaps?  Feel free to answer "offline" if you wish (it might be of limited interest, even among those who frequent apportionment discussions on this forum).

Maybe I don't know what the power mean is.

We know that if the divisors for Minnesota and New York were

(7*8) ^1/2  and  (26*27) ^1/2

resulted in Minnesota's quotient being approximately four greater than New York's.

But if the divisors were:

(7*8) ^{1/2.000006951}   and  (26*27) ^{1/2.000006951}

they would have been tied.

The value of p in the power mean of (# of seats a state already has been "awarded") and (# of seats a state is "going for") where Minnesota and New York would be tied for the last seat is p ≈ 0.0021.  Minnesota and Montana would be tied for the last seat at p ≈ 0.1017.  Minnesota oscillates between 7 and 8 seats several times as p increases from −∞ to ∞ (or decreases from ∞ to −∞).

Just indicating that I read this, but don't understand. I glanced at the power mean reference, and saw that there was a "cubic mean", and decided I could use any power.

Ranking methods and a fixed house size are fundamentally flawed. If Minnesota had lost the seat it would have increased its share of the USA population while simultaneously reducing it share of the USA representation.

All in the eye of the beholder.  Scholarly works will sometimes talk of divisor methods with a fixed house size (or at least fixed as far as that being the "key number" and not the ratio) as being the gold standard.  It doesn't have some of the paradoxes that largest remainder methods have.

Perhaps I should have written ranking methods with a fixed house size are fundamentally flawed.

Webster's method applied to the apportionment of the British Parliament is a quota violation generator, because the apportionment is between only four countries, and England is so dominant. A person moving from Scotland to Wales can cause Northern Ireland to gain or lose a seat. But this is only because the Parliament size was fixed at 500.

If you simply calculate a quota (total electorate/500), and divide that quota into the electorate of each country, you are calculate how many 500ths of the electorate is in each country.

A country with between n.5 and (n+1).5 /500ths of the electorate should have n+1 MPs. It does not matter that this results in 501 or 499, or even 502 or 498 MPs in Parliament.

[CityByTheValley]

The Census Bureau didn't publish how many numerical people each state was from displacing Minnesota or New York (assuming no other states gained/lost seats), so I calculated the 10 closest states to gaining and losing seats.

Gaining (Farthest to Closest)
Texas - 189,645
Florida - 171,561
Utah   - 136,978
Virginia - 111,635
Delaware - 88,205
Arizona - 79,509
West Virginia - 73,911
Idaho - 27,579
Ohio - 11,462
New York - 89

Losing (Closest to Farthest)
 Minnesota - -26
 Montana - -6,371
 Rhode Island - -19,127
 Oregon - -62,408
 Colorado - -72,445
 Alabama - -85,285
 Nebraska - -94,387
 North Carolina - -160,592
 South Carolina - -179,944
 Wisconsin   - -187,747

[Kevinstat]

I am almost certain that Webster's method would flip the seat from Minnesota to New York.

Actually, under Webster's method (or rather Major Fractions as used in 1910 and 1930, having a fixed House size rather than having a fixed ratio as was done in the beginning of our country's history (with a largest remainder method with a bunch of paradoxes in between fixed ratio and fixed House size divisor methods)), Minnesota would still have 8 seats but seats would have "moved" from Montana and Rhode Island to New York and Ohio.  Minnesota would have lost a seat under the Harmonic Mean ("Dean") divisor method to Idaho (which would have gained a 3rd seat), interestingly enough.

There is probably a value of p in the generalized or power mean between 0 (geometric mean; equal proportions) and 1 (arithmetic mean; major fractions) that would just flip a seat from Minnesota to New York, but I haven't tested that yet.  Montana got the second-to-last seat IRL so I can't be sure it wouldn't lose one before Minnesota without checking.

1 / 2.000006951 would tie the two.

The quotients using the square root for Minnesota and New York are only 4 apart while Montana is 4501 greater than Minnesota.

Using 1 / 2.000006951 would reduce the gap between Montana and Minnesota-New York to 4497.

I didn't understand quite what you were saying here.  Some slight weight to the geometric mean, perhaps?  Feel free to answer "offline" if you wish (it might be of limited interest, even among those who frequent apportionment discussions on this forum).

Maybe I don't know what the power mean is.

We know that if the divisors for Minnesota and New York were

(7* ^1/2  and  (26*27) ^1/2

resulted in Minnesota's quotient being approximately four greater than New York's.

But if the divisors were:

(7* ^{1/2.000006951}  and  (26*27) ^{1/2.000006951}

they would have been tied.

The value of p in the power mean of (# of seats a state already has been "awarded") and (# of seats a state is "going for") where Minnesota and New York would be tied for the last seat is p ≈ 0.0021.  Minnesota and Montana would be tied for the last seat at p ≈ 0.1017.  Minnesota oscillates between 7 and 8 seats several times as p increases from −∞ to ∞ (or decreases from ∞ to −∞).

Just indicating that I read this, but don't understand. I glanced at the power mean reference, and saw that there was a "cubic mean", and decided I could use any power.

Thanks for explaining where the 1 / 2.000006951 came from.

Being a math geek, it's tough for me to explain this well.  But the geometric mean of two numbers is the square root of the product of those numbers rather than the cube root or some other root not "just because," but because there are 2 numbers.  If you were taking the geometric mean of three numbers, you would take the cube root of the product, similar to how if you take the "regular" (arithmetic) mean of 3 numbers, you divide the sum of those numbers by 3 rather than by 2.  Doing what you did, just tweaking the exponent without changing anything else, is arguably problematic because, say, what if you changed the exponent to 1/200, or I'll be conservative and change it to 1/2.1 (the "2.1th root"), well, see below.

(26*27) ^1/2.1 ≈ 22.6672,

less than either of the two numbers.  Similarly,

(26*27) ^1/1.9 ≈ 31.4827,

greater than either of the two numbers.  Not much of a mean.  That's also the case for the hypothetical "means", calculated the same way, of 7 and 8 (≈ 6.7994 in the first case and ≈ 8.3195 in the second case), so you might think those oddities cancel out.  But they don't, really.

The ratio of the divisors for a state that already "has" 26 seats and going for a 27th and a state that already "has" 7 seats and is going for an 8th should be somewhere between 27/8 = 3.375 [27 and 8 is what the two divisors would be in the "greatest divisors", "D'Hondt" or "Jefferson" method] and 26/7 ≈ 3.7143 [27 and 7 is what the two divisors would be in the "smallest divisors" or "Adams" method].  And the ratios of the divisors for the arithmetic mean, 26.5/7.5 ≈ 3.5333, and the geometric mean,

[ (26*27) ^1/2 ] ÷ [ ( 7*8 ) ^1/2 ] ≈ 26.4953 ÷ 7.4833 ≈ 3.5406

both comply.  But [ (26*27) ^1/2.1 ] ÷ [ ( 7*8 ) ^1/2.1 ] ≈ 22.6672 ÷ 6.7994 ≈ 3.3337 (below that range)

and [ (26*27) ^1/1.9 ] ÷ [ ( 7*8 ) ^1/1.9 ] ≈ 31.4827 ÷ 8.3195 ≈ 3.7841 (above that range; I got lucky with the numbers I picked).

So hopefully you can see that just changing the root or power of the product doesn't really "work", at least not for something that can be described as a continuum of means or a continuum of divisor methods based on a continuum of means.

So, what does work?  Since the means we're taking are all of just two numbers (and adjacent integers at that, but that's beside the point), I'll adjust my definitions a bit accordingly.  For the generalized or power mean, you raise each number by an exponent p, you take the arithmetic mean (or what most people would think of when they think of mean or average) of the two resulting numbers, and then raise that average to the 1/p power (or, in other words, take the pth root of that average).

Stated a bit more eloquently, the power mean with exponent p is the pth root of the arithmetic mean of the p power of those two numbers (that explanation works pretty well for any number of numbers being averaged, actually).

For p = 1, the raising of exponents or taking of roots doesn't change anything, so you're left with the the arithmetic mean.  For p = -1, the negative 1st power (or negative first root) of a number is the reciprocal of that number, so you get the reciprocal of the arithmetic mean of the reciprocal of those two numbers.  The cubic mean you alluded to (p = 3, the cube root of the arithmetic mean of the cubes of those numbers) is actually on the "larger number side" (or the greatest divisors/D'Hondt/Jefferson "side") of the arithmetic mean (or major fractions/Webster), which itself is on the D'Hondt side of equal proportions, while you maybe thought it was somewhere on the smallest divisors/Adams "side" of equal proportions.

For p = 0, there's a dilemma, as the 0th root of a number is undefined.  But in some types of math you can use something called limits, or basically see what the result "approaches" (becomes increasingly and ultimately infinitesimally close to) as the independent variable "approaches" (becomes increasingly and ultimately infinitesimally close to) some problematic number.  Well, it turns out the the limit of the power mean (as definied above) as p approaches 0 is none other than the geometric mean (the nth root of the product of the n numbers, so the square root for our purposes).  A common convention seems to be to make an exception for what would otherwise be the definition of the power mean for p = 0, and to sub in the geometric mean (what the limit is as p approaches 0) in that case.  And the geometric mean shares various properties, which I won't get into (the Wikipedia article covers it a bit) with the other power means, so it seems to make sense.

Anyway, for the case of where the priority value for Minnesota's 8th seat would tie that for New York's 27th seat (for seat #435), it happens to be where the priority value for Minnesota's 8th seat is approximately

[(7^0.0021)*(8^0.0021)] ^1/0.0021

and the priority value for New York's 27th seat is approximately

[(26^0.0021)*(27^0.0021)] ^1/0.0021   .

For where the priority value for Minnesota's 8th seat would tie that for Montana's 2nd seat (also for seat #435), it's where the priority value for Minnesota's 8th seat is approximately

[(7^0.1017)*(8^0.1017)] ^1/0.1017

and the priority value for Montana's 2nd seat is approximately

[(1^0.1017)*(2^0.1017)] ^1/0.1017   .

So that's what I meant when I said that Minnesota and New York would be tied for the last seat is p ≈ 0.0021, and that Minnesota and Montana would be tied for the last seat at p ≈ 0.1017.

Does that explanation help at all?

[Kevinstat]

Webster's method applied to the apportionment of the British Parliament is a quota violation generator, because the apportionment is between only four countries, and England is so dominant. A person moving from Scotland to Wales can cause Northern Ireland to gain or lose a seat. But this is only because the Parliament size was fixed at 500.

Well, Paul LePage gained a Governor's seat in 2010 in part because some people (including me) voted two weeks early for Libby Mitchell and couldn't pivot to Elliot Cutler (the equivalent of the Andromeda Galaxy gaining a seat because people couldn't get visas to leave the U.S.S.R. for say... Denmark in time for the Local Group census, so how is your above example that different?  

[Born to Slay. Forced to Work.]

The Census Bureau didn't publish how many numerical people each state was from displacing Minnesota or New York (assuming no other states gained/lost seats), so I calculated the 10 closest states to gaining and losing seats.

Gaining (Farthest to Closest)
Texas - 189,645
Florida - 171,561
Utah   - 136,978
Virginia - 111,635
Delaware - 88,205
Arizona - 79,509
West Virginia - 73,911
Idaho - 27,579
Ohio - 11,462
New York - 89

Losing (Closest to Farthest)
 Minnesota - -26
 Montana - -6,371
 Rhode Island - -19,127
 Oregon - -62,408
 Colorado - -72,445
 Alabama - -85,285
 Nebraska - -94,387
 North Carolina - -160,592
 South Carolina - -179,944
 Wisconsin   - -187,747

Wait a damn minute, South Carolina could have lost a seat? All the time people in the state brag about having some of the fastest growing areas (Horry, Charleston, Greenville) and we could’ve lost one?