Some scattered thoughts regarding population deviation, calculation thereof, etc

[Kevinstat]

Equal population requirements, etc.:

The federal judicial standard for population equality (or the point at which the burden of proof in a lawsuit alleging malapportionment shifts from the plaintiff to the state or other apportioning body) for districts other than congressional districts seems to be that the population of the largest district can exceed that of the smallest by up to 10% of the ideal district population (so a district could be up to 7% overpopulated if no district was more than 3% underpopulated, or vice versa).  In Maine, for example, with 151 State House districts, one district could deviate from the ideal by 9.94% (10% × 150/151) in one direction or the other if the other 150 districts each had exactly the same population as each other (all deviating from the ideal by 0.06% in the other direction).  But when one is drawing districts, one doesn’t know what the population of not yet drawn districts will be, but if every district is within 5% of the ideal district population, you know the range will be within 10%.  So, keeping within a 5% deviation is a good rule of thumb, and that is sometimes colloquially treated as the standard.

When census population totals by municipality (or county in those states or parts of a state where counties are the chief unit of local government) come out, or when census estimates come out and I use projections comparing those totals to those of the previous census to project the population in the next census, I am interested in what side of multiples of 0.95 and 1.05 "quotas" (unrounded "ideal" decimal numbers of districts of a given type) they come out to.  This is particularly relevant in states where either state law, constitution or tradition respected across party lines prohibits a wanton disregard for local boundaries.  The integer quotas can also be significant in states where local government boundaries are largely respected, as it might raise significant eyebrows (or even be illegal) for a municipality or county with say, 1.99 quotas having two whole districts and part of a third, even though that could be done (even with a unit with 1.91 quotas).  But having just one whole district and most of a second might be seen as okay, even if not ideal, and perhaps the best move if a neighboring unit or conglomerate of units had excess population it needed to shed.  On the other hand, in a municipality or county with 2.01, having two whole districts and part of a third might seem okay (like if a neighboring potential district with good community of interest needed a bit more population to be brought within 5% of the ideal), while having only one whole district and one partial district (albeit one almost entirely in that municipality or county) would likely raise serious eyebrows, even though that could be done (even with a unit with 2.09 quotas).  I've heard people talk about a county having "n+1" districts (n whole districts and 1 partial districts), and one could think of either "1+1" districts or 2 being acceptable for a municipality or county in with between 1.9 and 2 quotas, and either 2 districts or "2+1" being acceptable for a county or municipality with between 2 and 2.1 quotas.  Between 1.05 and 1.9 quotas, the only acceptable option (in states with laws or accepted tradition as described above) is "1+1", and between 2.1 and 3.85 quotas, the only acceptable option is "2+1".

For congressional districts, the range where a local government unit could get away with having exactly a certain number of districts (all entirely within that unit, and with no part of that unit in another district) is very narrow, and in some instances you will have to divide a unit with less than a whole quota or the remainder of a larger unit, or give a larger unit one fewer "whole districts" than it is entitled and two partial districts, at least one of which and perhaps would be mostly in that unit in terms of its population).  But the same thing about "2+1" being odd in a unit with 1.99 quotas and "1+1" being odd in a unit with 2.01 quotas would apply.

Deep dive into how I calculate "quota" and deviation:

So how does one calculate a municipality's (or any other defined tract of territory's, like a hypothetical district's) "quota" as I’ve described it above?  How does one calculate deviation?  ([Municipal/Hypothetical District/whatever] Population) ÷ ((State Population) ÷ (# of districts)) = (([Municipal/Hypothetical District/whatever] Population) × (# of districts)) ÷ (State Population).  While the general practice seems to be to divide the state's population by the number of districts, round that figure to the nearest integer and then divide a proposed district’s population by that rounded figure to determine that district’s portion of an ideal district, and then subtract by 1 and then multiply by 100% to determine its percentage deviation from the ideal, I don't like to use rounded figures (even rounded to the nearest integer) in calculations.  Instead, I will multiply the population of the unit in question by the number of districts (both always integers, so the product will also always be an integer) and divide that product by the state’s population (also always an integer) to determine that unit’s decimal "quota" (and then subtract by 1 and then multiply by 100% to get the percentage deviation from an "ideal" district).  I like to think of myself as a mathematical purist.  Using the term "average district population rather than "ideal district population" would be more correct here, but the concept of an "ideal district population" is deeply ingrained in redistricting, and I'm not above using that term even while contemplating the less than ideal chopping up of persons.

I know persons can’t be divided in redistricting (yes, I'm talking to you, Trond), so if the population per State Representative is not an exact integer, the integer nearest that decimal number on either side can be thought of as the ideal population for a district, since the closest plan mathematically possible (even dividing homes, etc.) would have districts of both of those sizes.  But if you had a 4-person town divided into three districts for the Town Council, two of them with 1 person and the other one with 2 people, I would think of the 2-person district as being 50% overpopulated (dividing 2 by 4/3, or 1.33..., then subtracting 1 and then multiplying by 100%) and the two 1-person districts as each being 25% underpopulated or -25% overpopulated (dividing 2 by 4/3, or 1.33..., then subtracting 1 and then multiplying by 100%).  The 1.33... is what you'd be using as the average district population, whether you were aware of it or not.  You wouldn't just divide the district populations by the rounded population per councilor district, which would be 1 in this case.  And if rounding isn't appropriate in that case, I don't see why one needs to round the population per district, period.  In practice, the differences in the two calculations should be incredibly negligible.  Like in the 2010 Census, Maine's resident population was 1,328,361.  Dividing that by 151 (the number of State House districts in Maine) yields, rounded to the nearest four digits, 8,797.0927.  So pretty close to an integer.  But let's suppose Maine’s population was as close as possible to halfway between products of 151.  If it was 1,328,422, then the population per district would be 8,797.4967, which still rounds to 8,797.  Dividing the (fictitious) population of the entire state (the 1,328,422) by the rounded 8,797 figure yields (when rounding to the nearest four digits) 151.0085 districts under the traditional calculation compared to 151.0000 under mine.  To have a difference of (fully) 0.0001 State House quotas (so not just because a figure goes from #.####49 to #.####51), it would take a municipality with 15,582 people, slightly larger than Waterville in 2010.  And again, that's with a number deliberately chosen to have the maximum rounding needed to get to the nearest integer population per Representative.  So why do I even bother doing something different?  It's aesthetic, I guess.  Anyway, the figures I use on this site and elsewhere use my "mathematically pure" method of calculating a fractional quota or deviation.

I'll be curious to see what other's thoughts are on all this.

[jimrtex]

Equal population requirements, etc.:

The federal judicial standard for population equality (or the point at which the burden of proof in a lawsuit alleging malapportionment shifts from the plaintiff to the state or other apportioning body) for districts other than congressional districts seems to be that the population of the largest district can exceed that of the smallest by up to 10% of the ideal district population (so a district could be up to 7% overpopulated if no district was more than 3% underpopulated, or vice versa).  In Maine, for example, with 151 State House districts, one district could deviate from the ideal by 9.94% (10% × 150/151) in one direction or the other if the other 150 districts each had exactly the same population as each other (all deviating from the ideal by 0.06% in the other direction).  But when one is drawing districts, one doesn’t know what the population of not yet drawn districts will be, but if every district is within 5% of the ideal district population, you know the range will be within 10%.  So, keeping within a 5% deviation is a good rule of thumb, and that is sometimes colloquially treated as the standard.

The 10% standard largely comes from:

White v. Regester, 412 U.S. 755 (1973)

This particular version is preferred over those on Justia and Findlaw, because the table of districts and population deviation is presented in a better format. The other sites completely lose the formatting.

There are two parts to the SCOTUS decision. One concerns the use of multi-member districts, and the other concerns population deviation.

Incidentally, an All-Star Cast of Lawyers and personages. Mark White was SOS of Texas at the time, and later became governor. You will recall Leon Jaworski from Watergate. John Hill was AG at the time, and was later Chief Justice of the Texas Supreme Court. He also lost in a run for governor. David Richards was Ann Richard's husband at the time (they divorced in 1984). Jim Mattox was AG, and a congressman. Richards (Ann), Mattox, and White would face off in the 1990 Democratic Primary for governor.

The first legislative redistricting after the 1970 census was overturned by the Texas Supreme Court in Craddick v Smith. In that decision, a young Tom Craddick who was one of very few Republicans in the House, faced Preston Smith who was governor of Texas at the time. An older Craddick is seeking his 27th term this November. The one-man one-vote decisions had led some to believe that county boundaries could be (and must be ignored), and the legislature drew a district line down Craddick's residential street (in Midland).

The Texas Constitution calls for the apportionment of representatives among the counties of the state based on the quota determined by dividing the state population by the total number of representatives.

Sec. 26.  APPORTIONMENT OF MEMBERS OF HOUSE OF REPRESENTATIVES.  The members of the House of Representatives shall be apportioned among the several counties, according to the number of population in each, as nearly as may be, on a ratio obtained by dividing the population of the State, as ascertained by the most recent United States census, by the number of members of which the House is composed; provided, that whenever a single county has sufficient population to be entitled to a Representative, such county shall be formed into a separate Representative District, and when two or more counties are required to make up the ratio of representation, such counties shall be contiguous to each other; and when any one county has more than sufficient population to be entitled to one or more Representatives, such Representative or Representatives shall be apportioned to such county, and for any surplus of population it may be joined in a Representative District with any other contiguous county or counties.

The constitution requires that representatives be apportioned among one or more whole counties.

In particular, multi-county districts elect one representative, and single-county districts election one or more representatives (a single county, single representative district is a degenerate form of both types: a district with a single representative for N counties, where N=1; or a N-county district with one representative, where N is one).

The constitution also provides for floterial districts, where the surplus population (remnant above a whole number of representatives was treated as another county in a multi-county district.

In an ideal form, you might have two adjacent counties with populations of about 1.5 quotas, each would elect one representative, and then another representative would be elected from a district combining the two districts, with all voters participating.

This could also work under less stringent equal protection standards, where two counties entitled to 1.4 representatives might share a floterial district, while counties with 1.2 or 0.8 representatives might each have a district.

This might also work when one county had a population equivalent to 1.6 quotas and another to 1.4, since the larger county would only have 53% of the total electorate. It works less well when one county was equivalent to 3.5 quotas and some smaller counties had about 0.5 quotas. The larger county would have 87.5% of the votes, while only providing half of the population to earn the representative.

In the second case, deviation might be measured as 387.5%/350% - 1 = 10.7%; and 12.5%/50% - 1 = -75%. This is based on an assumption that the representative "represents" each constituent equally. You could come up with a different formula if you assume the representative from the floterial district provided less representation to all constituents.

At the time the constitution was written, most representatives were from counties that might be entitled to one or two representatives, with three being exceptionally large. Only on the frontier would there be multi-county districts. As the frontier advanced, some of these counties would gain their own representation.

Floterial districts were a simple way to handle those counties that fell intermediately between one and two representatives.

Floterial districts are fine for apportionment purposes, but not electoral purposes, because they violate equal protection.

In Craddick v Smith the Texas Supreme Court harmonized the Texas and US constitutions, by permitting a portion of a county that contained the surplus population for the county, to be treated as a whole county for apportionment purposes and electoral purposes.

For example, if the quota was 100,000 and a county had 350,000 persons, an area containing the surplus population of 50,000 could be set off and attached to adjacent counties (either whole or an area containing a surplus population). The remaining population of 300,000 would elect three representatives. It was later after at-large elections were largely banned, that the area with 300,000 persons would be divided into three districts. The Texas Supreme Court has ruled that this is a manner regulation, and not an apportionment.

After the state decision, the House was redistricted using quasi-floterial districts, multi-county single-member districts, and single-county multi-member districts was at issue in White v Regester It so happened that the largest districts had a deviation of 5.8% and the smallest a deviation of 4.1%. 5.8% minus 4.1% equals 9.9% which is less than 10%.

In the plan under consideration, there was one smaller county that had been divided. Bowie in the northeastern corner of the state was short of the quota, but adding either of its neighbors, Cass or Red River would place it beyond the quota. This a clear violation of the Texas Constitution, but required under equal protection. Under redistricting plans since then, there have generally been one exception to the Texas Constitution.

Because quasi-floterial districts are a technical violation of the Texas Constitution, they are avoided to the extent possible. Thus El Paso which is entitled to a little over 4.75 representatives, has 5 underpopulated districts, instead of 4 districts in the county, and one about 3/4 in the county and extending to the east (Hudspeth, Culberson, and beyond).

If you look at the deviations in the appendix to White v Regester you will see that they appear to follow a normal distribution (bell-shaped) curve. While two districts were almost to 6% deviation. Only 32 were over 3% deviation (there my be a distortion because the multi-district counties were treated as having all districts having the same population.

An analogous situation is when shooting an arrow at a target in an archery competition. You shoot towards the center of the bullseye, but score 10 points with any shot within the black ring. Judgement on whether the archer was shooting at the center of the target is not based on the extreme outlying shots.

The SCOTUS approach assumes that lawyers will ride down to the target in a golf cart, and stick the red-feathered arrows at one extreme, and the blue-feathered arrows at the opposite extreme. But so long as they are all in bounds, after a moving of the 10-point circle to fit it is OK.

Since that time, the distribution of deviations in Texas has begun to resemble more of a U-shape, as individual districts in larger counties can be manipulated. There isn't really much flexibility in the apportionment of districts among counties in Texas. This is particularly true in East Texas or around metropolitan areas, where there may be only one or two options. Deviations will still form bell curves.

Before White v Regester, the SCOTUS had in Abate v Mundt upheld the apportionment of supervisors in Rockland County, New York, where the apportionment of supervisors among the five towns was based on population. The largest deviation in population per supervisor was 4.8%, and the smallest -7.1%, for a total range of 11.9%. They had accepted the deviation because they believed it to be justified by the linkage between the supervisors and the towns (before the OMOV decisions, boards of supervisors in New York counties were comprised of the town supervisor from each town, along with supervisors from any city wards.

It was based on these two decisions that the burden-shifting threshold was set. An observation was made that in Abate v Mundt, the range of deviation had exceeded 10%, but the county had justified the deviation. While Texas was under 10%, they had not justified that range of deviation in the mind of the court.

Subsequently, after the 1990 Census, found the Rockland County system no longer justified. The deviation range had increased significantly, and the link to the towns had weakened. Suburbanization had weakened the recognition of towns, and people were more aware of their village (Rockland has 19) or school district. Rockland had also eliminated dual-hatting where a town official could serve on both the town and county governments. 1970 was just a fortunate quantization of town populations. Rockland now has some extremely ugly looking single-member districts.

Texas has continued with its apportionment scheme and has always met the 10% threshold.

It is likely that if Maine did produce a map with a single district with a deviation of 9.94%, and 150 districts with a deviation of -0.06% that it would violate the Maine Constitution that districts be as equal in population as prcticable.

Does Maine have any House districts that divide a town with a population less than a quota?

[jimrtex]

When census population totals by municipality (or county in those states or parts of a state where counties are the chief unit of local government) come out, or when census estimates come out and I use projections comparing those totals to those of the previous census to project the population in the next census, I am interested in what side of multiples of 0.95 and 1.05 "quotas" (unrounded "ideal" decimal numbers of districts of a given type) they come out to.  This is particularly relevant in states where either state law, constitution or tradition respected across party lines prohibits a wanton disregard for local boundaries.  The integer quotas can also be significant in states where local government boundaries are largely respected, as it might raise significant eyebrows (or even be illegal) for a municipality or county with say, 1.99 quotas having two whole districts and part of a third, even though that could be done (even with a unit with 1.91 quotas).  But having just one whole district and most of a second might be seen as okay, even if not ideal, and perhaps the best move if a neighboring unit or conglomerate of units had excess population it needed to shed.  On the other hand, in a municipality or county with 2.01, having two whole districts and part of a third might seem okay (like if a neighboring potential district with good community of interest needed a bit more population to be brought within 5% of the ideal), while having only one whole district and one partial district (albeit one almost entirely in that municipality or county) would likely raise serious eyebrows, even though that could be done (even with a unit with 2.09 quotas).  I've heard people talk about a county having "n+1" districts (n whole districts and 1 partial districts), and one could think of either "1+1" districts or 2 being acceptable for a municipality or county in with between 1.9 and 2 quotas, and either 2 districts or "2+1" being acceptable for a county or municipality with between 2 and 2.1 quotas.  Between 1.05 and 1.9 quotas, the only acceptable option (in states with laws or accepted tradition as described above) is "1+1", and between 2.1 and 3.85 quotas, the only acceptable option is "2+1".

The way that surplus is defined in Texas would seem to preclude this, since a county with 1.9 quotas has less than two full quotas.

There is one small county split in Texas due to Ellis (0.90) not having enough for a whole district, but with no neighbors with a small enough or small enough surplus to make Ellis whole.

But Dallas is entitled to 14.4 representatives, and has 14 districts with an average deviation of +2.86%.

If Dallas and Ellis were combined there could be 15 representatives for 15.3 quotas, an average deviation of +2.00%. One would have to squint a bit to let this to happen, since while Dallas has a surplus of 0.40 quotas, an area with a population of about 0.12 quotas would be attached to Ellis. This might still be preferred to ignoring the surplus entirely, since the creation of an additional quasi-floterial district would eliminate the division of Henderson.

[Kevinstat]

[Floterial districts] might also work when one county had a population equivalent to 1.6 quotas and another to 1.4, since the larger county would only have 53% of the total electorate. It works less well when one county was equivalent to 3.5 quotas and some smaller counties had about 0.5 quotas. The larger county would have 87.5% of the votes, while only providing half of the population to earn the representative.

In the second case, deviation might be measured as 387.5%/350% - 1 = 10.7%; and 12.5%/50% - 1 = -75%. This is based on an assumption that the representative "represents" each constituent equally. You could come up with a different formula if you assume the representative from the floterial district provided less representation to all constituents.

Ah yes, the component method, which I like as it both (a) has the deviations expressed in the same terms people are used to for states/areas without floterial districts (one could argue we should measure absolute and relative deviations in seats per capita, not people per seat, and I think one measuring method for floterial districts does just that) and (b) given how the deviations are expressed, expresses the (inverse of the) share of a representative in a given area as well as seemingly possible.

Floterial districts are fine for apportionment purposes, but not electoral purposes, because they violate equal protection.

Can you elaborate?  If all floterial districts are drawn exclusively in areas covered by non-floterial single- or multi-member districts (or other floterials that themselves are drawn exclusively in areas covered by non-floterial districts, or other floterials… but you eventually get to a non-floterial district everywhere), and no floterial district consists of part but not all of any "smaller-level" district for that same representative body (that might not even be key, but its aesthetically pleasing for me for there to be a sort of hierarchy where there are floterial districts), and all the non-floterial component districts are within the same deviation limits as for non-floterials when you use the component method, I don't see how it's any worse than normal multi-member districts or the mixed single- and multi-member districts many states use.  Granted, mixed single- and multi-member districts are bad enough (in my opinion), but I wouldn't single out floterials as being worse unless (and this is really bad) it was something like your example of a 3.5 quota county with its own 3-member district having a floterial district with an 0.5 quota county that doesn't have it's own district, and of course as you pointed out the component method would give such a plan a failing grade for population equality anyway.

[Kevinstat]

After [Craddick v Smith], the House was redistricted using quasi-floterial districts, multi-county single-member districts, and single-county multi-member districts was at issue in White v Regester It so happened that the largest districts had a deviation of 5.8% and the smallest a deviation of 4.1%. 5.8% minus 4.1% equals 9.9% which is less than 10%.

Interesting how certain burden-shifting standards (or what becomes assumed by semi-attuned observers to be the standard) can come about.

Because quasi-floterial districts are a technical violation of the Texas Constitution, they are avoided to the extent possible. Thus El Paso which is entitled to a little over 4.75 representatives, has 5 underpopulated districts, instead of 4 districts in the county, and one about 3/4 in the county and extending to the east (Hudspeth, Culberson, and beyond).

While the Senate district including El Paso County does extend to the east, which I know you've explained is preferable to the Trans-Pecos being in a district dominated by distant San Antonio or suburbs thereof, but considering how the ratio of State Representatives to State Senators in Texas is less than 5:1, having a State Senate district consist of five whole State House districts and part of a sixth (or maybe parts of multiple other districts, I don't know) seems weird to me.

If you look at the deviations in the appendix to White v Regester you will see that they appear to follow a normal distribution (bell-shaped) curve. While two districts were almost to 6% deviation. Only 32 were over 3% deviation (there my be a distortion because the multi-district counties were treated as having all districts having the same population.

An analogous situation is when shooting an arrow at a target in an archery competition. You shoot towards the center of the bullseye, but score 10 points with any shot within the black ring. Judgement on whether the archer was shooting at the center of the target is not based on the extreme outlying shots.

The SCOTUS approach assumes that lawyers will ride down to the target in a golf cart, and stick the red-feathered arrows at one extreme, and the blue-feathered arrows at the opposite extreme. But so long as they are all in bounds, after a moving of the 10-point circle to fit it is OK.

I love your comparisons like this.  Some more than others.  As a Democrat, I wasn't too keen on your impression of the Pennsylvania Supreme Court two years ago, but perhaps it was a fair one.

Before White v Regester, the SCOTUS had in Abate v Mundt upheld the apportionment of supervisors in Rockland County, New York, where the apportionment of supervisors among the five towns was based on population. The largest deviation in population per supervisor was 4.8%, and the smallest -7.1%, for a total range of 11.9%. They had accepted the deviation because they believed it to be justified by the linkage between the supervisors and the towns (before the OMOV decisions, boards of supervisors in New York counties were comprised of the town supervisor from each town, along with supervisors from any city wards.

It was based on these two decisions that the burden-shifting threshold was set. An observation was made that in Abate v Mundt, the range of deviation had exceeded 10%, but the county had justified the deviation. While Texas was under 10%, they had not justified that range of deviation in the mind of the court.

Again, interesting how legal standards (or what Joe Q. Public assumes them to be) come about.

[Kevinstat]

It is likely that if Maine did produce a map with a single district with a deviation of 9.94%, and 150 districts with a deviation of -0.06% that it would violate the Maine Constitution that districts be as equal in population as pr[a]cticable.

Well, in that specific case maybe but the whole part where the Maine Constitution says that House (and by later constitutional ditto) Senate districts "shall cross political subdivision lines the least number of times necessary to establish as nearly as practicable equally populated districts" seems to be interpreted as "balance fealty to political subdivisions with population equality, yada, yada, yada."  The Maine Supreme Judicial Court stated in 1983 and quoted itself in 2003 (in the "Conclusion" section of both decisions) that "the Legislature is not obligated to adopt the best apportionment plans conceivable, but only ones that comply with both the federal and the state constitutions."  You can perhaps file that under "duh", but it does suggest (particularly when you read the 1983 decision; the 2003 suit didn't allege malapportionment and was largely Green Party sour grapes, with one Penobscot Bay islander joining in; 1983 seems a bit different to me*) that nothing in the Maine Constitution about basic population equality should be taken as more strict than the federal standard.  The same would seem to go for division of towns with less than a whole quota.  For municipalities with a whole quota or more (for whichever type of district) it may be a different story, although I still wouldn't want to spend my savings hiring a lawyer to challenge a violation of those requirements and bank on getting my legal fees reimbursed as a result of my lawsuit succeeding.

*In 1983, from what I've pieced together, the Apportionment Commission pulled a Vermont (kind of) and didn't place the town of Harpswell in any House district, and the commission then, with no Republican members present, made several last-minute changes to fix that issue; enough Republicans voted for the plan for the 2/3 requirement to be met, but there was definitely some bitterness and some of the grievances were addressed in a 1986 state constitutional amendment (the odd year of the legislative biennium is always used in the law identifier, so that's why you see the 1985 there).

[Kevinstat]

Does Maine have any House districts that divide a town with a population less than a quota?

Yes, tons (well, maybe tons is an exaggeration but quite a few, and a few on the Senate side also).  Any municipality that is in two districts, neither one of which is entirely in the municipality has a population less than a quota.  One "minimal conglomerate" (I may not be using the right term there) of whole towns and whole districts on the House side has two districts entirely in Saco, the remainder of Saco in a district with Hollis (which doesn't border Saco) and part of Buxton connecting the two, the rest of Buxton in a district with Limington (which doesn't border Buxton) and parts of Standish (which connects the two) and Limerick, the rest of Standish (which is too big for a House district) being in a district entirely in Standish but the rest of Limerick being in with Newfield and Alfred (which doesn't border either Limerick or Newfield) and parts (majorities, it looks like) of Shapleigh (connecting Newfield to Alfred) and Parsonsfield, the rest of Shapleigh being in a district with Acton and Lebanon and the rest of Parsonsfield being in a district with Cornish, Baldwin, Sebago and Naples.  Again, of all these towns only Saco and Standish exceeded the quota.

On the Senate side, only Portland exceeds the quota (Lewiston in 2010 was in range of remaining coterminous with a Senate district and did so remain but had less than a whole quota), five organized municipalities in York and Cumberland Counties (so four not counting Portland) and two census-designated "county subdivisions" in unorganized territory are divided between Senate districts (one of the unorganized county subdivisions is split three ways).  It would be very difficult not to split a municipality other than Portland in the Portland area, unless you did something like take advantage of the water boundary between Portland and Cape Elizabeth and even then I don't know.  If you had gone with 31 Senate districts (you can have either 31, 33 or 35 State Senate seats under the Maine Constitution) in 2013 (when Maine redrew its non-congressional districts based on the 2010 census; we'll redraw everything except perhaps directly elected finance/budget committee in Aroostook and Knox counties in 2021), you could have put Portland in with Falmouth and some other town or towns beyond, South Portland in with Westbrook, Cape Elizabeth in with Scarborough and either OOB and Buxton or just Gorham, and probably gone on from there.  But either of the two options for a distirct with Cape Elizabeth and Scarborough will likely be too big for one Senate district out of 31 in 2020.  The same with the SoPo/Westbrook combo.  But the number of split cities or towns could have been lower.  Buxton is divided among both House and Senate districts, and not along the same line although at least the two lines don't cross each other so Buxton has "only" three ballot types rather than four.  Scarborough, with two whole House districts and part of a third, has both of its two whole House districts split between the same two Senate districts.  The remainder isn't split but does border the portion of Scarborough in the other Senate district.  So Scarborough has five ballot types, and last I knew both Scarborough and Buxton had only one polling place.

Statewide Senate District Map

Statewide House District Map

Additional district maps (including of individual districts) can be found here.

[jimrtex]

[Floterial districts] might also work when one county had a population equivalent to 1.6 quotas and another to 1.4, since the larger county would only have 53% of the total electorate. It works less well when one county was equivalent to 3.5 quotas and some smaller counties had about 0.5 quotas. The larger county would have 87.5% of the votes, while only providing half of the population to earn the representative.

In the second case, deviation might be measured as 387.5%/350% - 1 = 10.7%; and 12.5%/50% - 1 = -75%. This is based on an assumption that the representative "represents" each constituent equally. You could come up with a different formula if you assume the representative from the floterial district provided less representation to all constituents.

Ah yes, the component method, which I like as it both (a) has the deviations expressed in the same terms people are used to for states/areas without floterial districts (one could argue we should measure absolute and relative deviations in seats per capita, not people per seat, and I think one measuring method for floterial districts does just that) and (b) given how the deviations are expressed, expresses the (inverse of the) share of a representative in a given area as well as seemingly possible.

I was trying to remember (or re-invent) a method of measuring deviation in a case of layered districts. The component method has the advantage of (a) being additive; and (b) being more reflective of the concept of vote dilution. The conventional method of using population per-representative has the advantage of being directly observed when balancing district populations. For small deviations, 10% or so, it does not make a difference, particularly if the maximum range is being used.

Example:

1.05 - 0.95 = 10.00%

1/0.95 - 1/1.05 = 10.02%

Floterial districts are fine for apportionment purposes, but not electoral purposes, because they violate equal protection.

Can you elaborate?  If all floterial districts are drawn exclusively in areas covered by non-floterial single- or multi-member districts (or other floterials that themselves are drawn exclusively in areas covered by non-floterial districts, or other floterials… but you eventually get to a non-floterial district everywhere), and no floterial district consists of part but not all of any "smaller-level" district for that same representative body (that might not even be key, but its aesthetically pleasing for me for there to be a sort of hierarchy where there are floterial districts), and all the non-floterial component districts are within the same deviation limits as for non-floterials when you use the component method, I don't see how it's any worse than normal multi-member districts or the mixed single- and multi-member districts many states use.  Granted, mixed single- and multi-member districts are bad enough (in my opinion), but I wouldn't single out floterials as being worse unless (and this is really bad) it was something like your example of a 3.5 quota county with its own 3-member district having a floterial district with an 0.5 quota county that doesn't have it's own district, and of course as you pointed out the component method would give such a plan a failing grade for population equality anyway.

After Reynolds v Sims. a federal district court overturned a provision in the Texas Constitution that limited representation for larger counties. That provision had been added in 1936, and limited a county of 700,000 or less to 7 representatives. A county with more than 700,000 could have an additional representative for every 100,000 persons.

At the time, based on the 1930 Census, the quota would have been 38,831. The largest county, Harris, had a population of 359,328; and would have had 9 representatives, yet would need to double in population to begin to gain a seventh representative.

In any event, the legislature which had failed to reapportion in 1931, did not reapportion in 1941, even with the new provision. In 1948, the Legislative Redistricting Board was created as a backstop in case the legislature failed to reapportion. This provided an incentive for the legislature to redistrict as it did in in 1951 and 1961. There were only about a half-dozen floterial districts, as most rural counties had dropped below the threshold for a quota, due to urban growth and actual rural declines in population.

By 1961, Harris and Dallas had broken the 700,000 barrier, but were underrepresented, and Bexar, and now Tarrant were capped at seven. In 1964, in Kilgarlin v Martin a federal district court applied the newly discovered OMOV provisions of Reynolds v Sims to overturn the Texas Constitution's limit on larger counties.

The legislature redistricted in 1965, adding a total of 16 representatives to the four large counties:

Harris 19(+7)
Dallas 14(+5)
Bexar 10(+3)
Tarrant 8(+1)

This reduced the number of available seats for the remainder of Texas from 115 to 99. In 1961 there were five floterial districts where the dominant counties were Smith, Hidalgo, Cemeron, Grayson, and Potter. Effectively, other districts had to become 16.1% larger.

When the legislature redistricted in 1965 the need to reduce the number of representatives elsewhere resulted in creation of 11 floterial districts. For example, McLennan had three representatives in 1961, and was reduced to two representatives plus a floterial district, in effect expanding the area that was apportioned three representatives.

New floterial districts were created with dominant counties of Brazoria, McLennan, Bell, Travis, Nueces, Taylor, Ector, and Lubbock; while those involving Potter and Hidalgo were eliminated (these latter two were reduced from 2 and 3 and a little bit more, to exactly two and three). In all cases, what the court referred to as appurtenant counties had no other representation that through the floterial district.

In Kilgarlin v Hill the federal district court outlawed the use of floterial districts at least as practiced in Texas. This was a continuation of the earlier litigation over the restriction of representation for larger counties. The case also included a challenge to at-large elections. This decision was partially affirmed and partially reversed by the SCOTUS in Kilgarlin v Hill. By that time, John Hill had replaced Crawford Martin as SOS. The district court decision has a lot more detail than the SCOTUS decision.

The reasoning of the district court with regard to the floterial districts was curious. Nueces County had three representatives, and shared a floterial district with Kleberg.

The court reasoned that the floterial district with one representative for a population of about four quotas was overpopulated by about 300%, and also that the Nueces County districts were overpoulated because of a population well in excess of three representatives.

At the same time, it did note the vote dilution for voters in Kleberg (after all, the constitutional basis of Reynolds v Sims was vote dilution in violation of the equal protection clause). Deviation is a way to mathematically measure the dilution.

There is also a footnote that gives an example where a floterial district could be used, with two districts of roughly equal population sharing a third floterial district.

The district court ordered that for the 1966 election that the floterial districts be converted to at-large elections (e.g. four representatives were elected from Nueces and Kleberg).

The district court noted that Dallas and Bexar were still under-represented, and declined to address the variation among the other counties. That was the reasoning behind the SCOTUS reversal, which occurred in 1967, after the 1966 election.

The legislature redistricted again in 1967. They managed to add a representative in Dallas, and eliminated a district in West Texas, by shifting counties around, which presumably reduced the overall deviation.

The former floterials were treated in two ways. In about half the cases, the dominant count was divided, with the remnant attached to the appurtenant counties. I believe this was the first use of divided counties in Texas. In effect, this was where quasi-floterials were first employed.

In the other half, multi-county, multi-representative districts were used. Since single-member districts have been eliminated, this is no longer possible.

After the 1970 Census, the legislature redistricted, freely dividing smaller counties. 43 counties were divided, including Bexar, Harris, and Tarrant which had small areas attached to more rural districts.

There were only 14 multi-county, single representative districts, two single-county, single-representative districts (Denton and Gregg), and three multi-representative, single-county districts (Dallas(18 representatives), El Paso(5), and Travis(4).

Harris, which had been divided into three districts electing 7, 6, and 6 representatives in 1965, was divided into 6 districts, with 5 electing four representatives, and one electing three, plus a portion attached to other counties.

1972 Texas House Map

This is the map passed by the legislature, except that single-member districts were drawn in Harris, Dallas, and Bexar as a result of litigation. This was the map contested in White v Regester. Note that in White v Regester, the SCOTUS affirmed

Noteworthy:

(a) There are district numbers skipped and duplicated. It is hard to redistrict using a typewriter, copy of a map, and colored pencils, particularly with limited census maps and data.

(b) Small areas of Harris, Bexar, and Tarrant have small areas attached to outlying areas. Particularly interesting is Tarrant, with the central area (Ft.Worth, Arlington, HEB, etc. electing 9 representatives, and the outer then-rural areas being in a district with Parker. Bexar would have looked similar before single-member districts, with most of the county electing 11 representatives, while areas in the hills to the west and north being included in an district extending into the Hill Country.

(c) Smith, Brazoria, and Hidalgo have their surplus split.

(c) Numbering system for districts in Dallas and Bexar, using suffix letter. This was likely due to litigation forcing single-member districts.

(d) At the time, Grayson was more populous than Collin, Brazoria more populous than Fort Bend, and McLennan > Bell > Williamson.

Subsequent to this map, the federal district court ruled at-large elections invalid in the remaining multi-member counties, with the exception of Hidalgo(2): Tarrant(9), El Paso(4), Travis(4), Jefferson(3), Nueces(3), Lubbock(2), Galveston(2). This decision was on appeal to the SCOTUS, when the legislature drew district boundaries in these counties, and included Hidalgo. The SCOTUS remanded the case to the district court.

Meanwhile the VRA had been amended, placing Texas under pre-clearance. When Texas submitted its plan to the USDOJ, the USDOJ interposed objections to the districts in Jefferson, Nueces, and Tarrant.

Under the VRA, if a proposed change is blocked, the status quo remains in effect. Since this left the at-large districts in place, the case went back to the district court which had ruled the at-large districts unconstitutional. The parties in the lawsuit came to agreement with respect to Jefferson and Nueces, and the district court ordered those be used (a turefederal court is not subject to pre-clearance). They couldn't come to agreement as to Tarrant. The court ordered a plan proposed by the state, because it required fewer precinct changes before the upcoming 1976 primary.

After the election, the court reviewed the plan for Tarrant and adopted a version preferred by the plaintiffs. The reasoning of the court, was that the SCOTUS has adopted (see Chapman v Meier) a policy that a court, particularly a federal court, is subject to stricter limitations on deviation than a legislative body. The presumption is that a state legislature is more aware and understanding of the competing state interests.

So even though the deviation in the plans approved by the court for Jefferson and Nueces was greater, and the deviation was less than the SCOTUS had approved in White v Regester, the court decided that it was too great for them. In addition, the state alternative that had been used in 1976 had been proposed by the AG. And even though the legislature had passed a resolution in favor of the plan, that didn't count since the legislature hadn't actually passed a law.

I suspect the court may have been acting out because the SCOTUS had overturned their original decision on deviation.

[Kevinstat]

[Floterial districts] might also work when one county had a population equivalent to 1.6 quotas and another to 1.4, since the larger county would only have 53% of the total electorate. It works less well when one county was equivalent to 3.5 quotas and some smaller counties had about 0.5 quotas. The larger county would have 87.5% of the votes, while only providing half of the population to earn the representative.

In the second case, deviation might be measured as 387.5%/350% - 1 = 10.7%; and 12.5%/50% - 1 = -75%. This is based on an assumption that the representative "represents" each constituent equally. You could come up with a different formula if you assume the representative from the floterial district provided less representation to all constituents.

Ah yes, the component method, which I like as it both (a) has the deviations expressed in the same terms people are used to for states/areas without floterial districts (one could argue we should measure absolute and relative deviations in seats per capita, not people per seat, and I think one measuring method for floterial districts does just that) and (b) given how the deviations are expressed, expresses the (inverse of the) share of a representative in a given area as well as seemingly possible.

I was trying to remember (or re-invent) a method of measuring deviation in a case of layered districts. The component method has the advantage of (a) being additive; and (b) being more reflective of the concept of vote dilution. The conventional method of using population per-representative has the advantage of being directly observed when balancing district populations. For small deviations, 10% or so, it does not make a difference, particularly if the maximum range is being used.

Example:

1.05 - 0.95 = 10.00%

1/0.95 - 1/1.05 = 10.02%

Looking back over some old threads I had posted in, as I do from time to time, I realized that your above example wasn't quite the component method as it generally seems to be described because the deviations were expressed in terms of variance from the average share of a representative rather than population per representative.  The component method would have the deviation for the larger county as 350%/387.5% - 1 = -9.68%, and for the smaller county the deviation would be 50%/12.5% - 1 = 300%.  The smaller county has enough people for half a representative while only effectively having one eighth of one, and one half is 300% larger than one eighth.  This (component) method does, as I said above with slightly different wording, but not using a correct example, (a) express the deviations in the same terms people are used to for states/areas without floterial districts (particularly for single-member districts), that being in terms of population per representative and (b) given how the deviations are expressed, expresses variations from the average (inverse of the) share of a representative (the average population per representative) in a given area as well as seemingly possible.

But such a method as I just used, in addition to not being as reflective of the concept of vote dilution as you aptly pointed out, also seems less practical for an area with no non-floterial district of its own.

-----

Throwing in a third equation to your two examples above:

1.05/0.95 − 1 = (1/0.95)/(1/1.05) − 1 = 10.53%

If you're measuring as a percentage of the smaller figure (smaller in whatever measurement you're using, population per representative or share of a representative), the figure will be larger, and it gets substantially larger rather quickly.  I know the federal standards are not mentioned in such terms, but I just thought I'd throw that equation out there.  Maine state law for local election districts (like city wards) seems to use a 10% maximum variance in these terms [(largest district pop. ÷ smallest district pop.) − smallest district pop.], and the equivalent to a ±5% (from the average or "ideal") deviation range (or a 95% to 105% range) that is a good rule of thumb in the normal 10% case when you have a bunch of districts to draw... the equivalent good rule of thumb there for a maximum 1.1 largest district ÷ smallest district ratio would be a ±1/21 or approx. ±4.76% (or approx. 95.24% to 104.76%) range [1.0476/0.9524 − 1 ≈ 10%].  But if the number of voting districts in a given municipality is fairly small (there are just four wards in Augusta, and Augusta also has At-Large members of the Council and Board of Education but that wouldn't seem to affect the state law requirement), one may feel more free to go outside that range (or even the ±5% range) for a district than when there are 31 or 35 or 150 or 151 of them.  An Augusta ward could be off by as much as −6.98% (−3/43) or +7.32% (+3/41) and still have the largest ward exceed the smallest by only 10% if the other three wards had the exact same population as each other.