# How much can a US state grow by relative to the nation as a whole and still lose a seat?

Minnesota gained 7.6% over 2010's census, and the US as a whole gained 7.4%. New York gained 4.2%, which is the state that the seat would have gone to according to multiple reports if it had just 90 more people.

This strikes me as weird: if NYS gained 90+ more people than it did per the Census Minnesota would lose a seat despite growing slightly faster than the nation proportionally. Relative to the nation, what is the theoretical limit to how much a state's population can grow relative to the nation and still lose a seat in the US House?

• Why is this question being downvoted?? It is serious. Commented Jun 21, 2021 at 21:58
• "which is the state that the seat would have gone to" what seat? You haven't mentioned any seats until this point. Commented Jun 21, 2021 at 22:41
• A small state could double in population and still have a lower population growth than a large state. Commented Jun 21, 2021 at 22:54
• @Giter in a way voting systems are just mathematical problems. I don't really see why the theoretical limits would be uninteresting (or less interesting than historical edge cases). Knowing the theoretical limits would help predict which states are closer to losing seats or find which states only just held onto their seats.
– JJJ
Commented Jun 21, 2021 at 23:01
• @phoog No, lower absolute population growth, not in proportion. Wyoming (pop 580,000) vs California (pop 39,144,818) -- if Wyoming doubles in population and California only has a 1.5% increase, then California still gained 7,000 more individuals than Wyoming--Wyoming had lower absolute growth than California, even though a much larger relative growth. Commented Jun 23, 2021 at 12:56

Mathematically speaking this is a difficult question to address (similar to the knapsack problem) because one is trying to get a 'best fit' between a near-continuous variable — population percentage — and a fixed integer number of seats with a lower limit of one membership per state. But in simplest terms, New York did not lose a seat to Minnesota, and would not have gained a seat from Minnesota with 90 more people. Instead, they use a system called the Huntington-Hill method, which effectively amounts to placing all 435 House seats in a big pile, giving one House seat to each state, and dividing up the remaining seats one by one according to a 'priority' number calculated from the state's population and the geometric mean of the number of seats they currently have and the number of seats they would have if they earned the seat.

For example, New York has a census population of 20,215,751, while Minnesota has a census population of 5,706,494. Each starts with one House seat, so the initial priorities are as follows:

• New York: 20,215,751 / √( 1×2 ) = 14,294,694.6
• Minnesota: 5,709,752 / √( 1×2 ) = 4,037,404.4

New York will continue to have a higher priority than Minnesota until New York has been assigned four House seats:

• New York (with four House seats): 20,215,751 / √( 4×5 ) = 4,520,379.3
• New York (with five House seats): 20,215,751 / √( 5×6 ) = 3,690,874.3

So Minnesota cannot get a seat apportioned until New York has already gathered four. Minnesota will get a seat when its priority is higher than every other state's priority, and then its priority value will change to 2,330,996.5. The process will continue until all 435 seats are allocated. Small changes in population can affect the allocation of the last few seats somewhat chaotically — geometric means are funky things — so ups and downs are to be expected.

• Moreover, the OP's "this strikes me as weird" is in fact to be expected, as apportionment paradoxes are essentially unavoidable. Doesn't matter what you do, if you're trying to achieve the "obvious" fairness goals then you're going to get weird stuff going on in some cases. Commented Jun 22, 2021 at 0:38
• It would be useful to look at the last seat allocated (Minnesota) versus the first seat that would have been allocated if the House was one seat larger (New York). It was a squeaker. If New York's population was 89 (not 90!) people larger, the tossup would have gone in New York's favor. BTW, Minnesota's population per the 2020 census was 5709752 and New York's was 20215751. I'm not sure where you got your numbers. Commented Jun 22, 2021 at 13:13
• @DavidHammen: I may have grabbed the 2010 census numbers by accident; it was just for exemplification of the mathematical process, so I didn't put a whole lot of thought or effort into the google search. But when I get a chance today I'll redo the math with the numbers you've provided. Commented Jun 22, 2021 at 14:06
• @DavidHammen: And the problem with looking at the last seat is that we'd have to run through the calculations for every state for each of the 435 seats It isn't just NY vs MN. For all we know, New Mexico or Utah edged them both out for the 434th seat, setting up the NY/MN toss up for the last seat. Commented Jun 22, 2021 at 14:13
• @TedWrigley The choice for that last seat was amazingly close. Assuming that Minnesota's population was correct, if New York's population was low by 89 people that last seat should have gone to New York. Ohio (11462 people) and Idaho (11635 people) were also close, but nowhere near as close as New York. Commented Jun 22, 2021 at 14:20

Relative to the nation, what is the theoretical limit to how much a state's population can grow relative to the nation and still lose a seat in the US House?

The 2010 Census Briefs say the "The method of equal proportions has been used for apportionment after every census since 1940." It then goes on to detail the steps by which seats are apportioned:

1. Automatically assign the first 50 seats, as every state is guaranteed at least 1 Representative
2. Calculate a list of "priority values" using the formula PV(n) = (state apportionment population) / sqrt((next seat #) * (current seat #))
3. Sort priority values in descending order and assign seats from largest priority value to smallest

The Census Bureau provides a handy Excel file of the priority values for the 2010 census here and the 2020 census here.

It's easiest to get large growth percentages if you start small, so let's see what happens if we start with the smallest possible state that can have 2 reps.

The priority value for the 435th seat in 2010 was 710231, so you'd need a population of ~1,004,419 to get that second seat instead of Minnesota getting its eighth.

Just to get some idea of a possible bound, let's assume zero national population growth and that residents spread themselves evenly across some number of states n. I think the priority value of the 435th seat can then be calculated using (US population / n) / sqrt(ceil(385 / n) * (ceil(385/ n) - 1)). This yields a plot with a peak at 39 states, for which the priority value of the 435th seat is ~835661.5. This implies a population of roughly 1,181,803 to barely miss a seat, so a hypothetical state could increase its population by roughly 18.1% and still lose a seat.

As a warning, I haven't done the calculations to see if this difference would increase or decrease if the national population changes, but it at least shows what kind of difference is possible.

You can't just think about how much each state grew by compared to the nation, you also have to think about how fair the previous apportionment was. For example, consider this diagram showing two states.

The dashed line represents the 'fair' value, of population per seat. We can see that before this apportionment both states had populations per seat that were below the target value, ie. both states were overrepresented.

Both states grew in population and are now over the line, meaning that they are underrepresented. The first state grew more in population than the second, but because it started off being very overrepresented, most of its growth was only making it more fair, and it ends up being only slightly underrepresented. The second state didn't grow as much, but it ends up being more underrepresented. If there's a new seat up for grabs, it doesn't go to the state that grew the most, but the state that is the most underrepresented.

Of course reality is much more complex than this simplified example, and the fair target value changes with each apportionment, and the new seat doesn't actually just go to the state that is the most underrepresented, as the algorithm also takes into account how overrepresented the state would be if it received the seat. Ted Wrigley's answer gets into the details of how the US House of Representatives actually apportions its seats.

The other answers go into great detail about how representatives are apportioned. The more direct answer to the questions is that national growth is irrelevant to the apportionment process, growth comparisons between individual states are a bit better, but still not totally useful. While representatives are based on population, the actual method to apportion them is designed to have each person in each state represented as fair as possible with at least one representative per state, so the population growth is an indirect measure of how many representatives a state may get. There are states that could grow by double the national average and not see a change, and there are states that a few misplaced forms could affect the outcome drastically.

This all happens because the number of representatives is fixed regardless of population, rather than setting a number of representatives per population.