On December 21, 2010, the number of seats allocated to each state for the House of Representatives was announced. This allocation likely will determine representation to the House for the next five Congresses.
The Constitution requires that states be represented in the House of Representatives in accord with their population. It also requires that each state have at least one Representative, and that there be no more than one Representative for every 30,000 persons. For the 2010 apportionment, this could have meant a House of Representatives as small as 50 or as large as 10,306 Representatives.
Apportioning seats in the House of Representatives among the states in proportion to state population as required by the Constitution appears on the surface to be a simple task. In fact, however, the Constitution presented Congress with issues that provoked extended and recurring debate. How many Representatives should the House comprise? How populous should congressional districts be? What is to be done with the practically inevitable fractional entitlement to a House seat that results when the calculations of proportionality are made? How is fairness of apportionment to be best preserved? Apportioning the House can be viewed as a system with four main variables: (1) the size of the House, (2) the population of the states, (3) the number of states, and (4) the method of apportionment.
Over the years since the ratification of the Constitution, the number of Representatives has varied, but in 1941 Congress resolved the issue by fixing the size of the House at 435 members. How to apportion those 435 seats, however, continued to be an issue because of disagreement over how to handle fractional entitlements to a House seat in a way that both met constitutional and statutory requirements and minimized inequity.
The intuitive method of apportionment is to divide the United States population by 435 to obtain an average number of persons represented by a member of the House. This is sometimes called the ideal size congressional district. Then a state’s population is divided by the ideal size to determine the number of Representatives to be allocated to that state. The quotient will be a whole number plus a remainder—say 14.489326. What is Congress to do with the 0.489326 fractional entitlement? Does the state get 14 or 15 seats in the House? Does one discard the fractional entitlement? Does one round up at the arithmetic mean of the two whole numbers? At the geometric mean? At the harmonic mean? Congress has used, or at least considered, several methods over the years.
Every method Congress has used or considered has its advantages and disadvantages, and none has been exempt from criticism. Under current law, however, seats are apportioned using the equal proportions method, which is not without its critics. Some charge that the equal proportions method is biased toward small states. They urge Congress to adopt either the major fractions or the HamiltonVinton method as more equitable alternatives. A strong mathematical case can be made for either equal proportions or major fractions. Deciding between them is a policy matter based on whether minimizing the differences in district sizes in absolute terms (through major fractions) or proportional terms (through equal proportions) is most preferred by Congress.
On December 21, 2010, the number of seats allocated to each state for the House of Representatives was announced. This allocation likely will determine representation to the House for the next five Congresses.
The Constitution requires that states be represented in the House of Representatives in accord with their population. It also requires that each state have at least one Representative, and that there be no more than one Representative for every 30,000 persons. For the 2010 apportionment, this could have meant a House of Representatives as small as 50 or as large as 10,306 Representatives.
Apportioning seats in the House of Representatives among the states in proportion to state population as required by the Constitution appears on the surface to be a simple task. In fact, however, the Constitution presented Congress with issues that provoked extended and recurring debate. How many Representatives should the House comprise? How populous should congressional districts be? What is to be done with the practically inevitable fractional entitlement to a House seat that results when the calculations of proportionality are made? How is fairness of apportionment to be best preserved? Apportioning the House can be viewed as a system with four main variables: (1) the size of the House, (2) the population of the states, (3) the number of states, and (4) the method of apportionment.
Over the years since the ratification of the Constitution, the number of Representatives has varied, but in 1941 Congress resolved the issue by fixing the size of the House at 435 members. How to apportion those 435 seats, however, continued to be an issue because of disagreement over how to handle fractional entitlements to a House seat in a way that both met constitutional and statutory requirements and minimized inequity.
The intuitive method of apportionment is to divide the United States population by 435 to obtain an average number of persons represented by a member of the House. This is sometimes called the ideal size congressional district. Then a state's population is divided by the ideal size to determine the number of Representatives to be allocated to that state. The quotient will be a whole number plus a remainder—say 14.489326. What is Congress to do with the 0.489326 fractional entitlement? Does the state get 14 or 15 seats in the House? Does one discard the fractional entitlement? Does one round up at the arithmetic mean of the two whole numbers? At the geometric mean? At the harmonic mean? Congress has used, or at least considered, several methods over the years.
Every method Congress has used or considered has its advantages and disadvantages, and none has been exempt from criticism. Under current law, however, seats are apportioned using the equal proportions method, which is not without its critics. Some charge that the equal proportions method is biased toward small states. They urge Congress to adopt either the major fractions or the HamiltonVinton method as more equitable alternatives. A strong mathematical case can be made for either equal proportions or major fractions. Deciding between them is a policy matter based on whether minimizing the differences in district sizes in absolute terms (through major fractions) or proportional terms (through equal proportions) is most preferred by Congress.
One of the fundamental issues before the framers at the Constitutional Convention in 1787 was the allocation of representation in Congress between the smaller and larger states.^{2} The solution ultimately adopted, known as the Great (or Connecticut) Compromise, resolved the controversy by creating a bicameral Congress with states represented equally in the Senate, but in proportion to population in the House.
The Constitution provided the first apportionment of House seats: 65 Representatives were allocated among the states based on the framers' estimates of how seats might be apportioned following a census.^{3} House apportionments thereafter were to be based on Article 1, section 2, as modified by the Fourteenth Amendment:
Amendment XIV, section 2. Representatives shall be apportioned among the several States according to their respective numbers....
Article 1, section 2. The number of Representatives shall not exceed one for every thirty Thousand, but each State shall have at least one Representative....
From its beginning in 1789, Congress was faced with questions about how to apportion the House of Representatives—questions that the Constitution did not answer. How populous should a congressional district be on average? How many Representatives should the House comprise? Moreover, no matter how one specified the ideal population of a congressional district or the number of Representatives in the House, a state's ideal apportionment would, as a practical matter, always be either a fraction, or a whole number and a fraction—say, 14.489326. Thus, another question was whether that state would be apportioned 14 or 15 representatives? Consequently, these two major issues dominated the apportionment debate: how populous a congressional district ought to be (later recast as how large the House ought to be), and how to treat fractional entitlements to Representatives.^{4}
The questions of how populous a congressional district should be and how many Representatives should constitute the House have received little attention since the number of Representatives was last increased from 386 to 435 after the 1910 Census.^{5} The problem of fractional entitlement to Representatives, however, continued to be troublesome. Various methods were considered and some were tried, each raising questions of fundamental fairness. The issue of fairness could not be perfectly resolved: inevitable fractional entitlements and the requirement that each state have at least one representative lead to inevitable disparities among the states' average congressional district populations. Congress, which sought an apportionment method that would minimize those disparities, continued this debate until 1941, when it enacted the "equal proportions" method—the apportionment method still in use today (for a full explanation of this method, see below).
In light of the lengthy debate on apportionment, this report has four major purposes:
The process of apportioning seats in the House is constrained both constitutionally and statutorily. As noted previously, the Constitution defines both the maximum and minimum size of the House. There can be no fewer than one Representative per state, and no more than one for every 30,000 persons.^{6}
The Apportionment Act of 1941, in addition to specifying the apportionment method, sets the House size at 435, requires an apportionment every 10 years, and mandates administrative procedures for apportionment. The President is required to transmit to Congress "a statement showing the whole number of persons in each state" and the resulting seat allocation within one week after the opening of the first regular session of Congress following the census.^{7}
The Census Bureau has been assigned the responsibility of computing the apportionment. As a matter of practice, the Director of the Bureau reports the results of the apportionment at the end of December of the census year. Once received by Congress, the Clerk of the House of Representatives is charged with the duty of sending to the governor of each state a "certificate of the number of Representatives to which such state is entitled" within 15 days of receiving notice from the President.^{8}
An intuitive way to apportion the House is through simple rounding (a method never adopted by Congress). First, the U.S. apportionment population^{9} is divided by the total number of seats in the House (e.g., 309,183,463 divided by 435, in 2010) to identify the "ideal" sized congressional district (708,377 in 2010). Then, each state's population is divided by the "ideal" district population. In most cases this will result in a whole number and a fractional remainder, as noted earlier. Each state will definitely receive seats equal to the whole number, and the fractional remainders will either be rounded up or down (at the .5 "rounding point").
There are two fundamental problems with using simple rounding for apportionment, given a House of fixed size. First, it is possible that some state populations might be so small that they would be "entitled" to less than half a seat. Yet, the Constitution requires that every state must have at least one seat in the House. Thus, a method that relies entirely on rounding will not comply with the Constitution if there are states with very small populations. Second, even a method that assigns each state its constitutional minimum of one seat, and otherwise relies on rounding at the .5 rounding point, might require a "floating" House size because rounding at .5 could result in either fewer or more than 435 seats. Thus, this intuitive way to apportion fails because, by definition, it does not take into account the constitutional requirement that every state have at least one seat in the House and the statutory requirement that the House size be fixed at 435.
The current apportionment method (the method of equal proportions established by the 1941 act) satisfies the constitutional and statutory requirements. Although an equal proportions apportionment is not normally computed in the theoretical way described below, the method can be understood as a modification of the rounding scheme described above.
First, the "ideal" sized district is found (by dividing the apportionment population by 435) to serve as a "trial" divisor.
Then each state's apportionment population is divided by the "ideal" district size to determine its number of seats. Rather than rounding up any remainder of .5 or more, and down for less than .5, however, equal proportions rounds at the geometric mean of any two successive numbers. A geometric mean of two numbers is the square root of the product of the two numbers.^{10} If using the "ideal" sized district population as a divisor does not yield 435 seats, the divisor is adjusted upward or downward until rounding at the geometric mean will result in 435 seats.
For example, for the 2010 apportionment, the "ideal" size district of 708,377 had to be adjusted upward to between 709,063 and 710,231^{11} to produce a 435member House. Because the divisor is adjusted so that the total number of seats will equal 435, the problem of the "floating" House size is solved. The constitutional requirement of at least one seat for each state is met by assigning each state one seat automatically regardless of its population size.
Although the process of determining an apportionment through a series of trials using divisions near the "ideal" sized district as described above works, it is inefficient because it requires a series of calculations using different divisors until the 435 total is reached. Accordingly, the Census Bureau determines apportionment by computing a "priority" list of state claims to each seat in the House.
During the early 20^{th} century, Walter F. Willcox, a Cornell University mathematician, determined that if the rounding points used in an apportionment method are divided into each state's population (the mathematical equivalent of multiplying the population by the reciprocal of the rounding point), the resulting numbers can be ranked in a priority list for assigning seats in the House.^{12}
Such a priority list does not assume a fixed House size because it ranks each of the states' claims to seats in the House so that any size House can be chosen easily without the necessity of extensive recomputations.^{13}
The traditional method of constructing a priority list to apportion seats by the equal proportions method involves first computing the reciprocals^{14} of the geometric means (the "rounding points") between every pair of consecutive whole numbers (representing the seats to be apportioned). It is then possible to multiply by decimals rather than divide by fractions (the former being a considerably easier task). For example, the reciprocal of the geometric mean between 1 and 2 (1.41452) is 1/1.414452 or .70710678, which becomes the "multiplier" for the priorities for rounding to the second seat for each state. These reciprocals for all pairs (1 to 2, 2 to 3, 3 to 4, etc.) are computed for each "rounding point." They are then used as multipliers to construct the "priority list." Table 1, below, provides a list of multipliers used to calculate the "priority values" for each state in an equal proportions apportionment, allowing for the allocation of up to 60 seats to each state.
In order to construct the "priority list," each state's apportionment population is multiplied by each of the multipliers. The resulting products are ranked in order to show each state's claim to seats in the House. For example, (see Table 2, below) assume that there are three states in the Union (California, New York, and Florida) and that the House size is set at 30 Representatives. The first seat for each state is assigned by the Constitution; so the remaining 27 seats must be apportioned using the equal proportions formula. The 2010 apportionment populations for these states were 37,341,989 for California, 19,421,055 for New York, and 18,900,773 for Florida.
Once the priority values are computed, they are ranked with the highest value first. The resulting ranking is numbered and seats are assigned until the total is reached. By using the priority rankings instead of the rounding procedures described earlier in this paper under "The Formula in Theory," it is possible to see how an increase or decrease in the House size will affect the allocation of seats without the necessity of additional calculations.
Table 1. Multipliers for Determining Priority Values for Apportioning the House by the Equal Proportions Method
Seat Assignment 
Multiplier^{}^{}^{}^{}^{a}^{} 
Seat Assignment 
Multiplier^{}^{}^{}^{}^{a}^{} 
Seat Assignment 
Multiplier^{}^{}^{}^{}^{a}^{} 
1 
Constitution 
21 
0.04879500 
41 
0.02469324 
2 
0.70710678 
22 
0.04652421 
42 
0.02409813 
3 
0.40824829 
23 
0.04445542 
43 
0.02353104 
4 
0.28867513 
24 
0.04256283 
44 
0.02299002 
5 
0.22360680 
25 
0.04082483 
45 
0.02247333 
6 
0.18257419 
26 
0.03922323 
46 
0.02197935 
7 
0.15430335 
27 
0.03774257 
47 
0.02150662 
8 
0.13363062 
28 
0.03636965 
48 
0.02105380 
9 
0.11785113 
29 
0.03509312 
49 
0.02061965 
10 
0.10540926 
30 
0.03390318 
50 
0.02020305 
11 
0.09534626 
31 
0.03279129 
51 
0.01980295 
12 
0.08703883 
32 
0.03175003 
52 
0.01941839 
13 
0.08006408 
33 
0.03077287 
53 
0.01904848 
14 
0.07412493 
34 
0.02985407 
54 
0.01869241 
15 
0.06900656 
35 
0.02898855 
55 
0.01834940 
16 
0.06454972 
36 
0.02817181 
56 
0.01801875 
17 
0.06063391 
37 
0.02739983 
57 
0.01769981 
18 
0.05716620 
38 
0.02666904 
58 
0.01739196 
19 
0.05407381 
39 
0.02597622 
59 
0.01709464 
20 
0.05129892 
40 
0.02531848 
60 
0.01680732 
More specifically, for this example in Table 2, the computed priority values (column six) for each of the three states are ordered from largest to smallest. By constitutional provision, seats one to three are given to each state. The next determination is the fourth seat in the hypothesized chamber. California's claim to a second seat, based on its priority value, is 26,404,773.64 (0.70710681 x 37,341,989), while New York's claim to a second seat is 13,732,759.69 (0.70710681 x 19,421,055), and Florida's claim to a second seat is 13,364,864.76 (0.70710681 x 18,900,773). Based on the priority values, California has the highest claim for its second seat and is allocated the fourth seat in the hypothesized chamber.
Table 2. Priority Rankings for Assigning Thirty Seats in a Hypothetical ThreeState House Delegation
House Size 
State 
Seat Assignment 
Multiplier (M) 
Population (P) 
Priority Values (PxM) 
4 
CA 
2 
0.707106781 
26,404,773.64 

5 
CA 
3 
0.40824829 
37,341,989 
15,244,803.15 
6 
NY 
2 
0.707106781 
19,421,055 
13,732,759.69 
7 
FL 
2 
0.707106781 
18,900,773 
13,364,864.76 
8 
CA 
4 
0.288675135 
37,341,989 
10,779,703.70 
9 
CA 
5 
0.223606798 
37,341,989 
8,349,922.58 
10 
NY 
3 
0.40824829 
19,421,055 
7,928,612.50 
11 
FL 
3 
0.40824829 
18,900,773 
7,716,208.27 
12 
CA 
6 
0.182574186 
37,341,989 
6,817,683.24 
13 
CA 
7 
0.15430335 
37,341,989 
5,761,994.00 
14 
NY 
4 
0.288675135 
19,421,055 
5,606,375.67 
15 
FL 
4 
0.288675135 
18,900,773 
5,456,183.19 
16 
CA 
8 
0.133630621 
37,341,989 
4,990,033.18 
17 
CA 
9 
0.11785113 
37,341,989 
4,400,795.61 
18 
NY 
5 
0.223606798 
19,421,055 
4,342,679.92 
19 
FL 
5 
0.223606798 
18,900,773 
4,226,341.33 
20 
CA 
10 
0.105409255 
37,341,989 
3,936,191.25 
21 
CA 
11 
0.095346259 
37,341,989 
3,560,418.95 
22 
NY 
6 
0.18257419 
19,421,055 
3,545,783.30 
23 
FL 
6 
0.182574186 
18,900,773 
3,450,793.24 
24 
NY 
7 
0.15430335 
19,421,055 
2,996,733.85 
25 
FL 
7 
0.15430335 
18,900,773 
2,916,452.59 
26 
NY 
8 
0.133630621 
19,421,055 
2,595,247.64 
27 
FL 
8 
0.133630621 
18,900,773 
2,525,722.03 
28 
NY 
9 
0.11785113 
19,421,055 
2,288,793.28 
29 
FL 
9 
0.11785113 
18,900,773 
2,227,477.46 
30 
NY 
10 
0.10540926 
19,421,055 
2,047,158.95 
Notes: The Constitution requires that each state have at least one seat. Consequently, the first three seats assigned are not included in the table. Table prepared by CRS.
Next, the fifth seat's allocation is determined. California's claim to a third seat, based on the computed priority value, is 15,244,803.17 (0.40824829 x 37,341,989), while, as above, New York's claim to its second seat is 13,732,759.69 (0.70710681 x 19,421,055) and Florida's claim to its second seat is 13,364,864.76 (0.70710681 x 18,900,773). Again, California has a higher priority value, and is allocated its third seat, the fifth seat in the hypothesized chamber.
Next the sixth seat's allocation is determined in the same fashion. California's claim to a fourth seat, based on the computed priority value, is 10,779,703.70 (0.288675135 x 37,341,989), while, as above, New York's claim to its second seat is 13,732,759.69 (0.70710681 x 19,421,055) and Florida's claim to its second seat is 13,364,864.76 (0.70710681 x 18,900,773). As New York's priority value is higher than either California's or Florida's, it is allocated its second seat, the sixth seat in the hypothesized chamber.
Next, the seventh seat's allocation is determined. Again, California's claim to a fourth seat, based on the computed priority value, is 10,779,703.70 (0.288675135 x 37,341,989), while, having received its second seat, New York's claim to its third seat is 7,928,612.50(0.40824829 x 19,421,055) and Florida's claim to its second seat is 13,364,864.76 (0.70710681 x 18,900,773). As Florida's priority value is higher than either of the other states, Florida is, finally, allocated its second seat, the seventh seat in the hypothesized chamber. This same process is continued until all 30 seats in this hypothesized House are allocated to the three states.
From Table 2, then, we see that if the United States were made up of three states and the House size were to be set at 30 members, California would have 11 seats, New York would have 10, and Florida would have 9. Any other size House can be determined by picking points in the priority list and observing what the maximum size state delegation would be for each state.
A priority listing for all 50 states based on the 2010 Census is in the Appendix to this report. It shows priority rankings for the assignment of seats in a House ranging in size from 51 to 500 seats.
The equal proportions rule of rounding at the geometric mean results in differing rounding points, depending on which numbers are chosen. For example, the geometric mean between 1 and 2 is 1.4142, and the geometric mean between 49 and 50 is 49.49747. Table 3, below, shows the "rounding points" for assignments to the House using the equal proportions method for a state delegation size of up to 60. The rounding points are listed between each delegation size because they are the thresholds that must be passed in order for a state to be entitled to another seat. The table illustrates that, as the delegation size of a state increases, larger fractions are necessary to entitle the state to additional seats.
The fact that higher rounding points are necessary for states to obtain additional seats has led to charges that the equal proportions formula favors small states at the expense of large states. In Fair Representation, a 1982 study of congressional apportionment, authors M.L. Balinski and H.P. Young concluded that if "the intent is to eliminate any systematic advantage to either the small or the large, then only one method, first proposed by Daniel Webster in 1832, will do."^{15} This method, called the Webster method in Fair Representation, is also referred to as the major fractions method (major fractions uses the concept of the adjustable divisor as does equal proportions, but rounds at the arithmetic mean [.5] rather than the geometric mean.) Balinski and Young's conclusion in favor of major fractions, however, contradicts a report of the National Academy of Sciences (NAS) prepared at the request of House Speaker Nicholas Longworth in 1929. The NAS concluded that "the method of equal proportions is preferred by the committee because it satisfies ... [certain tests], and because it occupies mathematically a neutral position with respect to emphasis on larger and smaller states."^{16}
Size of Delegation 
Round Up At 
Size of Delegation 
Round Up At 
Size of Delegation 
Round Up At 
Size of Delegation 
Round Up At 
1 
1.41421 
16 
16.49242 
31 
31.49603 
46 
46.49731 
2 
2.44949 
17 
17.49286 
32 
32.49615 
47 
47.49737 
3 
3.46410 
18 
18.49324 
33 
33.49627 
48 
48.49742 
4 
4.47214 
19 
19.49359 
34 
34.49638 
49 
49.49747 
5 
5.47723 
20 
20.49390 
35 
35.49648 
50 
50.49752 
6 
6.48074 
21 
21.49419 
36 
36.49658 
51 
51.49757 
7 
7.48331 
22 
22.49444 
37 
37.49667 
52 
52.49762 
8 
8.48528 
23 
23.49468 
38 
38.49675 
53 
53.49766 
9 
9.48683 
24 
24.49490 
39 
39.49684 
54 
54.49771 
10 
10.48809 
25 
25.49510 
40 
40.49691 
55 
55.49775 
11 
11.48913 
26 
26.49528 
41 
41.49699 
56 
56.49779 
12 
12.49000 
27 
27.49545 
42 
42.49706 
57 
57.49783 
13 
13.49074 
28 
28.49561 
43 
43.49713 
58 
58.49786 
14 
14.49138 
29 
29.49576 
44 
44.49719 
59 
59.49790 
15 
15.49193 
30 
30.49590 
45 
45.49725 
60 
60.49793 
Notes: Any number between 709,063 and 710,231 divided into each state's 2010 population will produce a House size of 435 if rounded at these points, which are the geometric means of each pair of successive numbers. Table prepared by CRS.
A bill that would have changed the apportionment method to another formula called the "HamiltonVinton" method was introduced in 1981.^{17} The fundamental principle of the HamiltonVinton method is that it ranks fractional remainders. In order to reapportion the House using HamiltonVinton, each state's population would be divided by the "ideal" sized congressional district (309,183,463 divided by 435, in 2010, for an "ideal" district population of 708,377). Any state with fewer residents than the "ideal" sized district would receive a seat because the Constitution requires each state to have at least one House seat. The remaining states in most cases have a claim to a whole number and a fraction of a Representative. Each such state receives the whole number of seats it is entitled to. The fractional remainders are rankordered from highest to lowest until 435 seats are assigned. For the purpose of this analysis, we will concentrate on the differences between the equal proportions and major fractions methods because the HamiltonVinton method is subject to several mathematical anomalies.^{18}
Prior to the passage of the Apportionment Act of 1941 (2 U.S.C. 2(a)), the two contending methods considered by Congress were the equal proportions method (HillHuntington) and the method of major fractions (Webster). Each of the major competing methods—equal proportions (currently used) and major fractions—can be supported mathematically. Choosing between them is a policy decision, rather than a matter of conclusively proving that one approach is mathematically better than the other. A major fractions apportionment results in a House in which each citizen's share of his or her Representative is as equal as possible on an absolute basis. In the equal proportions apportionment now used, each citizen's share of his or her Representative is as equal as possible on a proportional basis. From a policy standpoint, a case can be made for either method of computing the apportionment of seats by arguing that one measure of fairness is preferable to the other.
As noted above, a major fractions apportionment results in a House in which each person's share of his or her Representative is as equal as possible on an absolute basis. As an example, in 2010, the state of North Carolina would have been assigned 14 seats under the major fractions method, and the state of Rhode Island would have received 1 seat. Under this allocation, there would have been 1.4636 Representatives per million for North Carolina residents and 0.9476 Representatives per million for Rhode Island residents. The absolute value^{19} of the difference between these two numbers is 0.5160.
Under the equal proportion method of assigning seats in 2010, North Carolina actually received 13 seats and Rhode Island 2. With 13 seats, North Carolina received 1.3590 Representatives for each million persons, and Rhode Island, with 2 seats, received 1.8953 Representatives per million persons. The absolute value of the difference between these two numbers is 0.5363. As this example shows, using the major fractions method produces a difference in the share of a Representative between the states that is smaller, in an absolute sense, than is the difference produced by the equal proportions method.
In addition, it can be argued that the major fractions minimization of absolute size differences among districts more closely reflects the "one person, one vote" principle established by the Supreme Court in its series of redistricting cases (Baker v. Carr, 369 U.S. 186 (1964) through Karcher v. Daggett, 462 U.S.725 (1983).^{20}
Although the "one person, one vote" rules have not been applied by the courts to apportioning seats among states, the method of major fractions can reduce the range between the smallest and largest district sizes more than the method of equal proportions—one of the measures that the courts have applied to withinstate redistricting cases. Although this range would have not changed in 2000 or 1990, if the method of major fractions had been used in 1980, the smallest average district size in the country would have been 399,592 (one of Nevada's two districts). With the method of equal proportions it was 393,345 (one of Montana's two districts). In both cases the largest district was 690,178 (South Dakota's single seat).^{21} Thus, in 1980, shifting from equal proportions to major fractions as a method of apportionment would have improved the 296,833 difference between the largest and smallest districts by 6,247 persons. It can be argued, because the equal proportions rounding points ascend as the number of seats increases, rather than staying at .5, that small states may be favored in seat assignments at the expense of large states. It is possible to demonstrate this by using simulation techniques.
The House has been reapportioned only 21 times since 1790. The equal proportions method has been used in five apportionments and the major fractions method in three. Eight apportionments do not provide sufficient historical information to enable policy makers to generalize about the impact of using differing methods. Computers, however, can enable reality to be simulated by using random numbers to test many different hypothetical situations. These techniques (such as the "Monte Carlo" simulation method) are a useful way to observe the behavior of systems when experience does not provide sufficient information to generalize about them.
Apportioning the House can be viewed as a system with four main variables: (1) the size of the House, (2) the population of the states,^{22} (3) the number of states,^{23} and (4) the method of apportionment.^{24} A 1984 exercise prepared for the Congressional Research Service (CRS) involving 1,000 simulated apportionments examined the results when two of these variables were changed—the method and the state populations. In order to further approximate reality, the state populations used in the apportionments were based on the Census Bureau's 1990 population projections available at that time. Each method was tested by computing 1,000 apportionments and tabulating the results by state. There was no discernible pattern by size of state in the results of the major fractions apportionment. The equal proportions exercise, however, showed that the smaller states were persistently advantaged.^{25}
Another way of evaluating the impact of a possible change in apportionment methods is to determine the odds of an outcome being different than the one produced by the current method—equal proportions. If equal proportions favors small states at the expense of large states, would switching to major fractions, a method that appears not to be influenced by the size of a state, increase the odds of the large states gaining additional representation? Based on the simulation model prepared for CRS, this appears to be true. The odds of any of the 23 largest states gaining an additional seat in any given apportionment range from a maximum of 13.4% of the time (California) to a low of .2% of the time (Alabama). The odds of any of the 21 multidistricted smaller states losing a seat range from a high of 17% (Montana, which then had two seats) to a low of 0% (Colorado), if major fractions were used instead of equal proportions.
In the aggregate, switching from equal proportions to major fractions "could be expected to shift zero seats about 37% of the time, to shift 1 seat about 49% of the time, 2 seats 12% of the time, and 3 seats 2% of the time (and 4 or more seats almost never), and, these shifts will always be from smaller states to larger states."^{26}
In summary, then, the method of major fractions minimizes the absolute differences in the share of a representative between congressional districts across states. In addition, it appears that the method of major fractions does not favor large or small states over the long term.
Support for the equal proportions formula primarily rests on the belief that minimizing the proportional differences among districts is more important than minimizing the absolute differences. Laurence Schmeckebier, a proponent of the equal proportions method, wrote in Congressional Apportionment in 1941, that
Mathematicians generally agree that the significant feature of a difference is its relation to the smaller number and not its absolute quantity. Thus the increase of 50 horsepower in the output of two engines would not be of any significance if one engine already yielded 10,000 horsepower, but it would double the efficiency of a plant of only 50 horsepower. It has been shown ... that the relative difference between two apportionments is always least if the method of equal proportions is used. Moreover, the method of equal proportions is the only one that uses relative differences, the methods of harmonic mean and major fraction being based on absolute differences. In addition, the method of equal proportions gives the smallest relative difference for both average population per district and individual share in a representative. No other method takes account of both these factors. Therefore the method of equal proportions gives the most equitable distribution of Representatives among the states.^{27}
An example using the North Carolina and Rhode Island 2010 populations illustrates the argument for proportional differences. The first step in making comparisons between the states is to standardize the figures in some fashion. One way of doing this is to express each state's representation in the House as a number of Representatives per million residents.^{28} The equal proportions formula assigned 13 seats to North Carolina and 2 to Rhode Island in 2010. If the major fractions method had been used, then 14 seats would have been assigned to North Carolina, and 1 would have been given to Rhode Island. Under this scenario, North Carolina has 1.4636 Representatives per million persons and Rhode Island has 0.9476 Representatives per million. The absolute difference between these numbers is 0.5160 and the proportional difference between the two states' Representatives per million is 54.45%. When 13 seats are assigned to North Carolina and 2 are assigned to Rhode Island (using equal proportions), North Carolina has 1.3590 Representatives per million and Rhode Island has 1.8953 Representatives per million. The absolute difference between these numbers is .0.5363 and the proportional difference is 39.46%.
Major fractions minimizes absolute differences, so in 2010, if this method had been required by law, North Carolina and Rhode Island would have received 14 and 1 seats respectively because the absolute difference (0.5160 Representatives per million) is smaller at 14 and 1 than it would be at 13 and 2 (0.5363). Equal proportions minimizes differences on a proportional basis, so it assigned 13 seats to North Carolina and 2 to Rhode Island because the proportional difference between a 13 and 2 allocation (39.46%) is smaller than would occur with a 14 and 1 assignment (54.45%).
The proportional difference versus absolute difference argument could also be cast in terms of the goal of "one person, one vote," as noted above. The courts' use of absolute difference measures in state redistricting cases may not necessarily be appropriate when applied to the apportionment of seats among states. The courts already recognize that the rules governing redistricting in state legislatures differ from those in congressional districting. If the "one person, one vote" standard were ever to be applied to apportionment of seats among states—a process that differs significantly from redistricting within states—proportional difference measures might be accepted as most appropriate.^{29}
If the choice between methods were judged to be a tossup with regard to which mathematical process is fairest, are there other representational goals that equal proportions meets that are, perhaps, appropriate to consider? One such goal might be the desirability of avoiding large districts, if possible. After the apportionment of 2010, five of the seven states with only one Representative (Alaska, Delaware, Montana, North Dakota, South Dakota, Vermont, and Wyoming) have relatively large land areas.^{30} The five Representatives of the larger states will serve 1.22% of the U.S. population, but also will represent 27% of the U.S. total land area.
Arguably, an apportionment method that would potentially reduce the number of very large (with respect to area size) districts would serve to increase representation in those states. Very large districts limit the opportunities of constituents to see their Representatives, may require more district based offices, and may require toll calls for telephone contact with the Representatives' district offices. Switching from equal proportions to major fractions may increase the number of states represented by only one member of Congress, although it is impossible to predict this outcome with any certainty using Census Bureau projections for 2025.^{31}
The table that follows contains the priority listing used in apportionment following the 2010 Census. Table A1 shows where each state ranked in the priority of seat assignments. The priority values listed beyond seat number 435 show which states would have gained additional representations if the House size had been increased.
Seat Sequence 
State 
Seat Number 
Priority Value 
51 
California 
2 
26,404,773.64 
52 
Texas 
2 
17,867,469.72 
53 
California 
3 
15,244,803.17 
54 
New York 
2 
13,732,759.69 
55 
Florida 
2 
13,364,864.76 
56 
California 
4 
10,779,703.70 
57 
Texas 
3 
10,315,788.45 
58 
Illinois 
2 
9,096,490.33 
59 
Pennsylvania 
2 
9,004,937.68 
60 
California 
5 
8,349,922.58 
61 
Ohio 
2 
8,180,161.26 
62 
New York 
3 
7,928,612.50 
63 
Florida 
3 
7,716,208.27 
64 
Texas 
4 
7,294,363.97 
65 
Michigan 
2 
7,008,577.96 
66 
Georgia 
2 
6,878,427.88 
67 
California 
6 
6,817,683.24 
68 
North Carolina 
2 
6,764,028.61 
69 
New Jersey 
2 
6,227,843.68 
70 
California 
7 
5,761,994.00 
71 
Virginia 
2 
5,683,537.63 
72 
Texas 
5 
5,650,190.03 
73 
New York 
4 
5,606,375.67 
74 
Florida 
4 
5,456,183.19 
75 
Illinois 
3 
5,251,861.14 
76 
Pennsylvania 
3 
5,199,003.20 
77 
California 
8 
4,990,033.18 
78 
Washington 
2 
4,775,353.02 
79 
Ohio 
3 
4,722,818.31 
80 
Massachusetts 
2 
4,638,368.75 
81 
Texas 
6 
4,613,360.84 
82 
Indiana 
2 
4,597,312.72 
83 
Arizona 
2 
4,534,463.66 
84 
Tennessee 
2 
4,508,110.49 
85 
California 
9 
4,400,795.61 
86 
New York 
5 
4,342,679.92 
87 
Missouri 
2 
4,250,756.86 
88 
Florida 
5 
4,226,341.33 
89 
Maryland 
2 
4,094,098.06 
90 
Michigan 
3 
4,046,404.37 
91 
Wisconsin 
2 
4,029,257.07 
92 
Georgia 
3 
3,971,262.19 
93 
California 
10 
3,936,191.25 
94 
North Carolina 
3 
3,905,213.74 
95 
Texas 
7 
3,899,001.55 
96 
Minnesota 
2 
3,758,186.98 
97 
Illinois 
4 
3,713,626.63 
98 
Pennsylvania 
4 
3,676,250.41 
99 
New Jersey 
3 
3,595,647.23 
100 
Colorado 
2 
3,567,304.21 
101 
California 
11 
3,560,418.95 
102 
New York 
6 
3,545,783.30 
103 
Florida 
6 
3,450,793.24 
104 
Alabama 
2 
3,396,221.14 
105 
Texas 
8 
3,376,634.39 
106 
Ohio 
4 
3,339,536.85 
107 
South Carolina 
2 
3,285,200.43 
108 
Virginia 
3 
3,281,391.98 
109 
California 
12 
3,250,202.96 
110 
Louisiana 
2 
3,220,137.41 
111 
Kentucky 
2 
3,076,343.00 
112 
New York 
7 
2,996,733.85 
113 
California 
13 
2,989,751.88 
114 
Texas 
9 
2,977,911.62 
115 
Florida 
7 
2,916,452.59 
116 
Illinois 
5 
2,876,562.82 
117 
Michigan 
4 
2,861,239.97 
118 
Pennsylvania 
5 
2,847,611.33 
119 
Georgia 
4 
2,808,106.42 
120 
California 
14 
2,767,972.38 
121 
North Carolina 
4 
2,761,403.12 
122 
Washington 
3 
2,757,051.35 
123 
Oregon 
2 
2,721,375.40 
124 
Massachusetts 
3 
2,677,963.45 
125 
Texas 
10 
2,663,525.12 
126 
Oklahoma 
2 
2,662,173.59 
127 
Indiana 
3 
2,654,259.74 
128 
Arizona 
3 
2,617,973.81 
129 
Tennessee 
3 
2,602,758.81 
130 
New York 
8 
2,595,247.64 
131 
Ohio 
5 
2,586,794.12 
132 
California 
15 
2,576,842.05 
133 
New Jersey 
4 
2,542,506.54 
134 
Connecticut 
2 
2,532,593.45 
135 
Florida 
8 
2,525,722.03 
136 
Missouri 
3 
2,454,175.62 
137 
California 
16 
2,410,415.03 
138 
Texas 
11 
2,409,249.13 
139 
Maryland 
3 
2,363,728.62 
140 
Illinois 
6 
2,348,703.70 
141 
Wisconsin 
3 
2,326,292.66 
142 
Pennsylvania 
6 
2,325,064.91 
143 
Virginia 
4 
2,320,294.52 
144 
New York 
9 
2,288,793.28 
145 
California 
17 
2,264,190.66 
146 
Florida 
9 
2,227,477.46 
147 
Michigan 
5 
2,216,306.95 
148 
Texas 
12 
2,199,333.49 
149 
Georgia 
5 
2,175,149.88 
150 
Minnesota 
3 
2,169,790.27 
151 
Iowa 
2 
2,159,353.50 
152 
North Carolina 
5 
2,138,973.66 
153 
California 
18 
2,134,699.43 
154 
Ohio 
6 
2,112,108.56 
155 
Mississippi 
2 
2,105,933.70 
156 
Arkansas 
2 
2,069,156.37 
157 
Colorado 
3 
2,059,584.05 
158 
New York 
10 
2,047,158.95 
159 
Kansas 
2 
2,025,021.59 
160 
Texas 
13 
2,023,092.56 
161 
California 
19 
2,019,223.51 
162 
Florida 
10 
1,992,316.41 
163 
Illinois 
7 
1,985,016.93 
164 
New Jersey 
5 
1,969,417.09 
165 
Pennsylvania 
7 
1,965,038.50 
166 
Alabama 
3 
1,960,809.19 
167 
Utah 
2 
1,959,226.72 
168 
Washington 
4 
1,949,529.71 
169 
Nevada 
2 
1,915,857.74 
170 
California 
20 
1,915,603.62 
171 
South Carolina 
3 
1,896,711.35 
172 
Massachusetts 
4 
1,893,606.11 
173 
Indiana 
4 
1,876,845.06 
174 
Texas 
14 
1,873,019.76 
175 
Louisiana 
3 
1,859,147.20 
176 
New York 
11 
1,851,724.94 
177 
Arizona 
4 
1,851,187.04 
178 
Tennessee 
4 
1,840,428.40 
179 
California 
21 
1,822,102.49 
180 
Michigan 
6 
1,809,607.05 
181 
Florida 
11 
1,802,118.00 
182 
Virginia 
5 
1,797,292.41 
183 
Ohio 
7 
1,785,057.53 
184 
Kentucky 
3 
1,776,127.46 
185 
Georgia 
6 
1,776,002.44 
186 
North Carolina 
6 
1,746,464.68 
187 
Texas 
15 
1,743,686.50 
188 
California 
22 
1,737,306.56 
189 
Missouri 
4 
1,735,364.22 
190 
Illinois 
8 
1,719,075.09 
191 
Pennsylvania 
8 
1,701,773.26 
192 
New York 
12 
1,690,385.87 
193 
Maryland 
4 
1,671,408.53 
194 
California 
23 
1,660,053.90 
195 
Florida 
12 
1,645,101.13 
196 
Wisconsin 
4 
1,644,937.31 
197 
Texas 
16 
1,631,069.37 
198 
New Jersey 
6 
1,608,022.32 
199 
California 
24 
1,589,380.60 
200 
Oregon 
3 
1,571,186.82 
201 
New York 
13 
1,554,928.84 
202 
Ohio 
8 
1,545,905.17 
203 
Oklahoma 
3 
1,537,006.64 
204 
Minnesota 
4 
1,534,273.41 
205 
Texas 
17 
1,532,122.89 
206 
Michigan 
7 
1,529,397.10 
207 
California 
25 
1,524,480.32 
208 
Illinois 
9 
1,516,081.72 
209 
Florida 
13 
1,513,272.94 
210 
Washington 
5 
1,510,099.22 
211 
Georgia 
7 
1,500,996.02 
212 
Pennsylvania 
9 
1,500,822.95 
213 
North Carolina 
7 
1,476,032.05 
214 
Virginia 
6 
1,467,483.11 
215 
Massachusetts 
5 
1,466,780.99 
216 
California 
26 
1,464,673.31 
217 
Connecticut 
3 
1,462,193.51 
218 
New Mexico 
2 
1,461,782.76 
219 
Colorado 
4 
1,456,345.85 
220 
Indiana 
5 
1,453,797.93 
221 
Texas 
18 
1,444,499.31 
222 
New York 
14 
1,439,584.37 
223 
Arizona 
5 
1,433,923.31 
224 
Tennessee 
5 
1,425,589.71 
225 
California 
27 
1,409,382.55 
226 
Florida 
14 
1,401,018.51 
227 
Alabama 
4 
1,386,501.48 
228 
Texas 
19 
1,366,359.56 
229 
Ohio 
9 
1,363,360.21 
230 
New Jersey 
7 
1,359,026.91 
231 
California 
28 
1,358,115.01 
232 
Illinois 
10 
1,356,024.72 
233 
Missouri 
5 
1,344,207.35 
234 
Pennsylvania 
10 
1,342,376.85 
235 
South Carolina 
4 
1,341,177.46 
236 
New York 
15 
1,340,180.12 
237 
Michigan 
8 
1,324,496.74 
238 
West Virginia 
2 
1,315,087.80 
239 
Louisiana 
4 
1,314,615.59 
240 
California 
29 
1,310,446.91 
241 
Florida 
15 
1,304,277.25 
242 
Georgia 
8 
1,299,900.68 
243 
Texas 
20 
1,296,242.49 
244 
Nebraska 
2 
1,295,295.88 
245 
Maryland 
5 
1,294,667.48 
246 
North Carolina 
8 
1,278,281.25 
247 
Wisconsin 
5 
1,274,162.96 
248 
California 
30 
1,266,011.99 
249 
Kentucky 
4 
1,255,911.77 
250 
New York 
16 
1,253,623.71 
251 
Iowa 
3 
1,246,703.32 
252 
Virginia 
7 
1,240,249.59 
253 
Washington 
6 
1,232,990.85 
254 
Texas 
21 
1,232,972.55 
255 
Illinois 
11 
1,226,570.51 
256 
California 
31 
1,224,492.06 
257 
Florida 
16 
1,220,039.65 
258 
Ohio 
10 
1,219,426.44 
259 
Mississippi 
3 
1,215,861.39 
260 
Pennsylvania 
11 
1,214,225.55 
261 
Massachusetts 
6 
1,197,621.66 
262 
Arkansas 
3 
1,194,627.99 
263 
Minnesota 
5 
1,188,443.07 
264 
Indiana 
6 
1,187,021.04 
265 
California 
32 
1,185,609.34 
266 
New York 
17 
1,177,574.43 
267 
New Jersey 
8 
1,176,951.83 
268 
Texas 
22 
1,175,593.20 
269 
Arizona 
6 
1,170,793.48 
270 
Kansas 
3 
1,169,146.76 
271 
Michigan 
9 
1,168,096.33 
272 
Tennessee 
6 
1,163,989.12 
273 
California 
33 
1,149,120.28 
274 
Georgia 
9 
1,146,404.65 
275 
Florida 
17 
1,146,027.70 
276 
Utah 
3 
1,131,160.07 
277 
Colorado 
5 
1,128,080.64 
278 
North Carolina 
9 
1,127,338.10 
279 
Texas 
23 
1,123,318.20 
280 
Illinois 
12 
1,119,700.56 
281 
California 
34 
1,114,810.42 
282 
Idaho 
2 
1,112,631.81 
283 
Oregon 
4 
1,110,996.86 
284 
New York 
18 
1,110,227.82 
285 
Pennsylvania 
12 
1,108,431.21 
286 
Nevada 
3 
1,106,120.98 
287 
Ohio 
11 
1,103,012.72 
288 
Missouri 
6 
1,097,540.70 
289 
Oklahoma 
4 
1,086,827.82 
290 
California 
35 
1,082,490.18 
291 
Florida 
18 
1,080,485.28 
292 
Texas 
24 
1,075,495.29 
293 
Virginia 
8 
1,074,087.65 
294 
Alabama 
5 
1,073,979.42 
295 
Maryland 
6 
1,057,091.57 
296 
California 
36 
1,051,991.36 
297 
New York 
19 
1,050,170.38 
298 
Michigan 
10 
1,044,777.12 
299 
Washington 
7 
1,042,067.46 
300 
Wisconsin 
6 
1,040,349.70 
301 
South Carolina 
5 
1,038,871.59 
302 
New Jersey 
9 
1,037,973.95 
303 
Connecticut 
4 
1,033,926.94 
304 
Texas 
25 
1,031,578.85 
305 
Illinois 
13 
1,029,974.71 
306 
Georgia 
10 
1,025,375.49 
307 
California 
37 
1,023,164.20 
308 
Florida 
19 
1,022,036.75 
309 
Pennsylvania 
13 
1,019,608.41 
310 
Louisiana 
5 
1,018,296.86 
311 
Massachusetts 
7 
1,012,175.04 
312 
North Carolina 
10 
1,008,321.85 
313 
Ohio 
12 
1,006,908.25 
314 
Indiana 
7 
1,003,215.88 
315 
New York 
20 
996,279.10 
316 
California 
38 
995,874.90 
317 
Texas 
26 
991,108.90 
318 
Arizona 
7 
989,501.09 
319 
Tennessee 
7 
983,750.36 
320 
Kentucky 
5 
972,825.08 
321 
Minnesota 
6 
970,359.71 
322 
California 
39 
970,003.60 
323 
Florida 
20 
969,589.20 
324 
Hawaii 
2 
966,517.39 
325 
Texas 
27 
953,694.98 
326 
Illinois 
14 
953,571.29 
327 
New York 
21 
947,650.45 
328 
Virginia 
9 
947,256.27 
329 
California 
40 
945,442.56 
330 
Michigan 
11 
945,036.46 
331 
Pennsylvania 
14 
943,973.96 
332 
Maine 
2 
942,625.67 
333 
New Hampshire 
2 
934,402.72 
334 
New Jersey 
10 
928,392.12 
335 
Missouri 
7 
927,591.19 
336 
Georgia 
11 
927,487.03 
337 
Ohio 
13 
926,220.87 
338 
Florida 
21 
922,263.29 
339 
California 
41 
922,094.69 
340 
Colorado 
6 
921,073.99 
341 
Texas 
28 
919,003.48 
342 
North Carolina 
11 
912,061.43 
343 
New York 
22 
903,549.25 
344 
Washington 
8 
902,456.89 
345 
California 
42 
899,872.28 
346 
Maryland 
7 
893,405.44 
347 
Illinois 
15 
887,726.56 
348 
Texas 
29 
886,747.63 
349 
Iowa 
4 
881,552.37 
350 
Florida 
22 
879,343.54 
351 
Wisconsin 
7 
879,255.98 
352 
Pennsylvania 
15 
878,791.93 
353 
California 
43 
878,695.85 
354 
Alabama 
6 
876,900.53 
355 
Massachusetts 
8 
876,569.30 
356 
Indiana 
8 
868,810.44 
357 
New York 
23 
863,371.20 
358 
Michigan 
12 
862,696.31 
359 
Oregon 
5 
860,574.46 
360 
Mississippi 
4 
859,743.83 
361 
California 
44 
858,493.24 
362 
Ohio 
14 
857,513.90 
363 
Arizona 
8 
856,933.08 
364 
Texas 
30 
856,679.60 
365 
Tennessee 
8 
851,952.80 
366 
South Carolina 
6 
848,235.10 
367 
Virginia 
10 
847,251.77 
368 
Georgia 
12 
846,675.94 
369 
Arkansas 
4 
844,729.55 
370 
New Mexico 
3 
843,960.67 
371 
Oklahoma 
5 
841,853.21 
372 
Florida 
23 
840,241.85 
373 
New Jersey 
11 
839,762.27 
374 
California 
45 
839,198.79 
375 
North Carolina 
12 
832,594.37 
376 
Louisiana 
6 
831,435.90 
377 
Illinois 
16 
830,392.16 
378 
Texas 
31 
828,584.07 
379 
Kansas 
4 
826,711.60 
380 
New York 
24 
826,615.00 
381 
Pennsylvania 
16 
822,034.58 
382 
California 
46 
820,752.61 
383 
Minnesota 
7 
820,103.63 
384 
Florida 
24 
804,470.32 
385 
Missouri 
8 
803,317.54 
386 
California 
47 
803,099.96 
387 
Texas 
32 
802,273.07 
388 
Connecticut 
5 
800,876.37 
389 
Utah 
4 
799,850.96 
390 
Ohio 
15 
798,302.00 
391 
Washington 
9 
795,892.17 
392 
Kentucky 
6 
794,308.35 
393 
Michigan 
13 
793,565.19 
394 
New York 
25 
792,861.25 
395 
California 
48 
786,190.69 
396 
Nevada 
4 
782,145.65 
397 
Illinois 
17 
780,017.61 
398 
Georgia 
13 
778,828.59 
399 
Colorado 
7 
778,449.60 
400 
Texas 
33 
777,581.81 
401 
Maryland 
8 
773,711.81 
402 
Massachusetts 
9 
773,061.46 
403 
Pennsylvania 
17 
772,167.04 
404 
Florida 
25 
771,620.83 
405 
California 
49 
769,978.84 
406 
New Jersey 
12 
766,594.56 
407 
Virginia 
11 
766,368.06 
408 
Indiana 
9 
766,218.79 
409 
North Carolina 
13 
765,875.43 
410 
New York 
26 
761,756.45 
411 
Wisconsin 
8 
761,458.01 
412 
West Virginia 
3 
759,266.29 
413 
Arizona 
9 
755,743.94 
414 
California 
50 
754,422.10 
415 
Texas 
34 
754,365.16 
416 
Tennessee 
9 
751,351.75 
417 
Nebraska 
3 
747,839.42 
418 
Ohio 
16 
746,743.14 
419 
Rhode Island 
2 
746,172.31 
420 
Florida 
26 
741,349.31 
421 
Alabama 
7 
741,116.21 
422 
California 
51 
739,481.57 
423 
Illinois 
18 
735,407.66 
424 
Michigan 
14 
734,698.60 
425 
New York 
27 
733,000.49 
426 
Texas 
35 
732,494.84 
427 
Pennsylvania 
18 
728,006.06 
428 
California 
52 
725,121.34 
429 
Georgia 
14 
721,055.17 
430 
South Carolina 
7 
716,889.51 
431 
Florida 
27 
713,363.71 
432 
Washington 
10 
711,867.60 
433 
Texas 
36 
711,857.03 
434 
California 
53 
711,308.24 
435 
Minnesota 
8 
710,230.58 
Last seat assigned by current law 

436 
North Carolina 
14 
709,062.86 
437 
Missouri 
9 
708,459.48 
438 
New York 
28 
706,336.94 
439 
New Jersey 
13 
705,164.44 
440 
Montana 
2 
703,158.30 
441 
Louisiana 
7 
702,691.59 
442 
Oregon 
6 
702,656.11 
443 
Ohio 
17 
701,443.04 
444 
Virginia 
12 
699,595.12 
445 
California 
54 
698,011.59 
446 
Illinois 
19 
695,626.00 
447 
Texas 
37 
692,350.39 
448 
Massachusetts 
10 
691,447.19 
449 
Pennsylvania 
19 
688,624.80 
450 
Florida 
28 
687,414.47 
451 
Oklahoma 
6 
687,370.27 
452 
Indiana 
10 
685,326.92 
453 
California 
55 
685,202.95 
454 
Michigan 
15 
683,967.17 
455 
Iowa 
5 
682,847.53 
456 
Maryland 
9 
682,349.68 
457 
New York 
29 
681,545.42 
458 
Arizona 
10 
675,957.93 
459 
Colorado 
8 
674,157.13 
460 
Texas 
38 
673,884.38 
461 
California 
56 
672,855.94 
462 
Tennessee 
10 
672,029.43 
463 
Wisconsin 
9 
671,542.85 
464 
Kentucky 
7 
671,313.08 
465 
Georgia 
15 
671,265.83 
466 
Mississippi 
5 
665,954.71 
467 
Florida 
29 
663,287.10 
468 
Ohio 
18 
661,326.84 
469 
California 
57 
660,946.04 
470 
North Carolina 
15 
660,101.60 
471 
Illinois 
20 
659,928.77 
472 
New York 
30 
658,435.43 
473 
Texas 
39 
656,377.90 
474 
Arkansas 
5 
654,324.70 
475 
Connecticut 
6 
653,912.82 
476 
Pennsylvania 
20 
653,286.84 
477 
New Jersey 
14 
652,855.41 
478 
California 
58 
649,450.45 
479 
Washington 
11 
643,908.47 
480 
Virginia 
13 
643,533.91 
481 
Idaho 
3 
642,378.28 
482 
Alabama 
8 
641,825.47 
483 
Florida 
30 
640,796.22 
484 
Kansas 
5 
640,368.05 
485 
Michigan 
16 
639,792.71 
486 
Texas 
40 
639,758.04 
487 
California 
59 
638,347.91 
488 
Delaware 
2 
637,016.24 
489 
New York 
31 
636,841.48 
490 
Missouri 
10 
633,665.42 
491 
Georgia 
16 
627,911.69 
492 
Illinois 
21 
627,717.47 
493 
California 
60 
627,618.61 
494 
Minnesota 
9 
626,364.50 
495 
Ohio 
19 
625,552.57 
496 
Massachusetts 
11 
625,437.52 
497 
Texas 
41 
623,959.11 
498 
Pennsylvania 
21 
621,399.74 
499 
South Carolina 
8 
620,844.52 
500 
Indiana 
11 
619,901.52 
Notes: Prepared by CRS.
1. 
A similar, previous CRS report was authored by [author name scrubbed], who retired in 2005. While the current report is modified by the current author, Mr. Huckabee's contribution, in a large part, remains. Of course, any errors that may appear are due solely to the current author. 
2. 
In part, this debate over the apportionment of power in the early years of this country came from the 10year experience with the unicameral congress provided for under the Articles of Confederation, which assigned one vote to each state delegation in Congress. For a thorough discussion, see Charles A. Kromkowski, Recreating the American Republic, (Cambridge University Press, Cambridge, U.K., 2002), esp., pp. 261307. 
3. 
A major controversy occurred even over the fixed, shortterm apportionment of seats among the delegates at the Constitutional Convention. See Kromkowski, pp. 287294. 
4. 
Thomas Jefferson recommended discarding the fractions. Daniel Webster and others argued that Jefferson's method was unconstitutional because it discriminated against small states. Webster argued that an additional Representative should be awarded to a state if the fractional entitlement was 0.5 or greater—a method that decreased the size of the house by 17 members in 1832. Congress subsequently used a "fixed ratio" method proposed by Rep. Samuel Vinton following the census of 1850 through 1900, but this method led to the paradox that Alabama lost a seat even though the size of the House was increased in 1880. Subsequently, mathematician W.F. Willcox proposed the "major fractions" method, which was used following the census of 1910. This method, too, had its critics; and in 1921 Harvard mathematician E.V. Huntington proposed the "equal proportions" method and developed formulas and computational tables for all of the other known, mathematically valid apportionment methods. A committee of the National Academy of Sciences conducted an analysis of each of those methods—smallest divisors, harmonic mean, equal proportions, major fractions, and greatest divisors—and recommended that Congress adopt Huntington's equal proportions method. For a review of this history, see U.S. Congress, House, Committee on Post Office and Civil Service, Subcommittee on Census and Statistics, The Decennial Population Census and Congressional Apportionment, 91^{st} Congress, 2^{nd} session. H. Report 911314 (Washington: GPO, 1970), Appendix B, pp. 1518. Also, see Michel L. Balinski and H. Peyton Young, Fair Representation, 2^{nd} edition, (Brookings Institution Press, Washington, 2001). 
5. 
Article I, Section 2 defines both the maximum and minimum size of the House, but the actual House size is set by law. There can be no fewer than one Representative per state, and no more than one for every 30,000 persons. Thus, the House after 2010 could be as small as 50 and as large as 10,306 Representatives. 
6. 
The actual language in of Article 1, section 2 pertaining to this minimum size reads as follows: "The number of Representatives shall not exceed one for every thirty Thousand, but each State shall have at least one Representative." This clause is sometime misread to be a requirement that districts can be no larger than 30,000 persons, rather than as it should be read, as a minimumsize population requirement. 
7. 
55 Stat. 761. (1941) Sec. 22 (a). [Codified in 2 U.S.C. 2(a).] In other words, after the 2010 Census, this report is due in January 2010. Interestingly, while the Constitution requires a census every ten years, it does not require that an apportionment of seats to the House of Representatives must occur. This became a statutory requirement with the passage of the Apportionment Act of 1941. 
8. 
Ibid., Sec. 22 (b). 
9. 
The apportionment population is the resident population of the 50 states. It excludes the population of the District of Columbia and U.S. territories and possessions, but since 1970, excepting 1980, it has included the overseas federal and military employees and their families. 
10. 
The geometric mean of 1 and 2 is the square root of 2, which is 1.4142. The geometric mean of 2 and 3 is the square root of 6, which is 2.4495. Geometric means are computed for determining the rounding points for the size of any state's delegation size. Equal proportions rounds at the geometric mean (which varies) rather than the arithmetic mean (which is always halfway between any pair of numbers). Thus, a state which would be entitled to 10.4871 seats before rounding will be rounded down to 10 because the geometric mean of 10 and 11 is 10.4881. The rationale for choosing the geometric mean rather than the arithmetic mean as the rounding point is discussed below in the section analyzing the equal proportions and major fractions formulas. 
11. 
Any number in this range divided into each state's population and rounded at the geometric mean will produce a 435seat House, with the provision that each state receives at least one seat. 
12. 
U.S. Congress, House Committee on Post Office and Civil Service, Subcommittee on the Census and Statistics, The Decennial Population Census and Congressional Apportionment, 91^{st} Congress, 2^{nd} session, H. Report 911814, (Washington: GPO, 1970), p. 16. 
13. 
The 435 limit on the size of the House is a statutory requirement. The House size was first fixed at 435 by the Apportionment Act of 1911 (37 Stat. 13). The Apportionment Act of 1929 (46 Stat. 26), as amended by the Apportionment Act of 1941 (54 Stat. 162), provided for "automatic reapportionment" rather than requiring the Congress to pass a new apportionment law each decade. This requirement to "automatically reapportion" every 10 years was needed because the Constitution, ironically, while requiring a census every 10 years makes no such requirement for apportionments. Thus, the fact that no apportionment was carried out after the 1920 census in no way violated the Constitution or any statutory requirement at the time. By authority of section 9 of PL 85508 (72 Stat. 345) and section 8 of PL 863 (73 Stat. 8), which admitted Alaska and Hawaii to statehood, the House size was temporarily increased to 437 until the reapportionment resulting from the 1960 Census when it returned to 435. 
14. 
A reciprocal of a number is that number divided into one. 
15. 
Fair Representation, pp. 34. (An earlier major work in this field was written by Laurence F. Schmeckebier, Congressional Apportionment (Washington: The Brookings Institution, 1941). Daniel Webster proposed this method to overcome the largestate bias in Jefferson's discarded fractions method. Webster's method was used three times, in the reapportionments following the 1840, 1910, and 1930 Censuses. 
16. 
"Report of the National Academy of Sciences Committee on Apportionment" in The Decennial Population Census and Congressional Apportionment, Appendix C, p. 21. 
17. 
H.R. 1990, 97^{th} Congress was introduced by Representative Floyd Fithian and was cosponsored by 10 other members of the Indiana delegation. Changing to the HamiltonVinton method would have kept Indiana from losing a seat. Hearings were held, but no further action was taken on the measure. U.S. Congress, House Committee on Post Office and Civil Service, Subcommittee on Census and Population, Census Activities and the Decennial Census, hearing, 97^{th} Cong., 1^{st} sess., June 11, 1981, (Washington: GPO, 1981). Since that time no other bill has been introduced to change the formula. 
18. 
The HamiltonVinton method (used after the 18501900 censuses) is subject to the "Alabama paradox" and various other population paradoxes. The Alabama paradox was so named in 1880 when it was discovered that Alabama would have lost a seat in the House if the size of the House had been increased from 299 to 300. Another paradox, known as the population paradox, has been variously described, but in its modern form (with a fixed size House) it works in this way: two states may gain population from one census to the next. State "A," which is gaining population at a rate faster than state "B," may lose a seat to state "B." There are other paradoxes of this type. HamiltonVinton is subject to them, whereas equal proportions and major fractions are not. 
19. 
The absolute value of a number is its magnitude without regard to its sign. For example, the absolute value of 8 is 8. The absolute value of the expression (42) is 2. The absolute value of the expression (24) is also 2. 
20. 
Major fractions best conforms to the spirit of these decisions if the population discrepancy is measured on an absolute basis, as the courts have done in the recent past. The Supreme Court has never applied its "one person, one vote" rule to apportioning seats of the House of Representatives among states (as opposed to redistricting within states). Thus, no established rule of law is being violated. Arguably, no apportionment method can meet the "one person, one vote" standard required by the Supreme Court for districts within states unless the size of the House is increased significantly (thereby making districts less populous). 
21. 
Nevada had two seats with a population of 799,184. Montana was assigned two seats with a population of 786,690. South Dakota's single seat was required by the Constitution (with a population of 690,178). The vast majority of the districts based on the 1980 census (323 of them) fell within the range of 501,000 to 530,000). 
22. 
For varying the definition of the population, see CRS Report RS22124, Potential House Apportionment Following the 2010 Census Based on Census Bureau Population Projections, by [author name scrubbed] (pdf), and, CRS Report R41636, Apportioning Seats in the U.S. House of Representatives Using the 2010 Estimated Citizen Population: 2012 , by [author name scrubbed]. 
23. 
For information on the impact of adding states, see CRS Report RS22579, District of Columbia Representation: Effect on House Apportionment, by [author name scrubbed], and CRS Report R41113, Puerto Rican Statehood: Effects on House Apportionment, by [author name scrubbed]. 
24. 
See CRS Report R41382, The House of Representatives Apportionment Formula: An Analysis of Proposals for Change and Their Impact on States, by [author name scrubbed]. 
25. 
H.P. Young and M.L. Balinski, Evaluation of Apportionment Methods, Prepared under a contract for the Congressional Research Service of the Library of Congress. (Contract No. CRS8415), Sept. 30, 1984. This document is available to Members of Congress and congressional staff from the author of this report. Comparing equal proportions and major fractions using the state populations from the 19 actual censuses taken since 1790, reveals that the small states would have been favored 3.4% of the time if equal proportions had been used for all the apportionments. Major fractions would have also favored small states, in these cases, but only .06% of the time. See Fair Representation, p. 78. 
26. 
Young and Balinski, Evaluation of Apportionment Methods, p. 13. 
27. 
Schmeckebier, Congressional Apportionment, p. 60. 
28. 
Representatives per million is computed by dividing the number of Representatives assigned to the state by the state's population (which gives the number of Representatives per person) and then multiplying the resulting dividend by 1,000,000. 
29. 
Montana argued in Federal court in 1991 and 1992 that the equal proportions formula violated the Constitution because it "does not achieve the greatest possible equality in number of individuals per Representative" Department of Commerce v. Montana 503 U.S. 442 (1992). Writing for a unanimous court, Justice Stevens however, noted that absolute and relative differences in district sizes are identical when considering deviations in district populations within states, but they are different when comparing district populations among states. Justice Stevens noted, however, "although common sense" supports a test requiring a "good faith effort to achieve precise mathematical equality within each State ... the constraints imposed by Article I, §2, itself make that goal illusory for the nation as a whole." He concluded "that Congress had ample power to enact the statutory procedure in 1941 and to apply the method of equal proportions after the 1990 census." 
30. 
The total area of the U.S. is 3,795,951 square miles. The area and (rank) among all states in area for the seven single district states in this scenario are as follows: Alaska−664,988 (1), Delaware−2,489 (49), Montana−147,039 (4), North Dakota−70,698 (19), South Dakota−77,116 (17), Vermont−9,616 (45), Wyoming−97,812 (10), Source: U.S. Department of Commerce, U.S. Census Bureau, Statistical Abstract of the United States, 2010, (Washington: GPO, 2010), Table 346: Land and Water Area of the States and Other Entities: 2008, p. 215. 
31. 
U.S. Census Bureau, Projections of the Total Population of States: 19952025, Series A, http://www.census.gov/population/projections/stpjpop.txt. If the major fractions method had been used to apportion the House in 2010, the number of states with a single Representative would have increased by one, from seven to eight, with the addition of Rhode Island. 