The U.S. House of Representatives Apportionment Formula in Theory and Practice

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 Hamilton-Vinton 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.

The U.S. House of Representatives Apportionment Formula in Theory and Practice

August 2, 2013 (R41357)

Contents

Summary

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 Hamilton-Vinton 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.


The U.S. House of Representatives Apportionment Formula in Theory and Practice

The U.S. House of Representatives Apportionment Formula in Theory and Practice1

Introduction

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 re-cast 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:

  • 1. summarize the constitutional and statutory requirements governing apportionment;
  • 2. explain how the current apportionment formula works in theory and in practice;
  • 3. summarize challenges to it on grounds of inequity; and
  • 4. explain the reasoning underlying the choice of the equal proportions method over its chief alternative, the method of major fractions.

Constitutional and Statutory Requirements

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

The Apportionment Formula

The Formula in Theory

An intuitive way to apportion the House is through simple rounding (a method never adopted by Congress). First, the U.S. apportionment population9 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 710,23111 to produce a 435-member 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.

The Formula in Practice: Deriving the Apportionment from a Table of "Priority Values"

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 20th 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 re-computations.13

The traditional method of constructing a priority list to apportion seats by the equal proportions method involves first computing the reciprocals14 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

Multipliera

Seat Assignment

Multipliera

Seat Assignment

Multipliera

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

a. Table by CRS, calculated by determining the reciprocal of the geometric mean of successive numbers, 1/√n (n-1), where "n" is the number of seats to be allocated to the state.

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 Three-State House Delegation

House Size

State

Seat Assignment

Multiplier (M)

Population (P)

Priority Values (PxM)

4

CA

2

0.707106781

37,341,989

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.

Challenges to the Current Formula

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

Table 3. Rounding Points for Assigning Seats Using
the Equal Proportions Method of Apportionment

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 "Hamilton-Vinton" method was introduced in 1981.17 The fundamental principle of the Hamilton-Vinton method is that it ranks fractional remainders. In order to reapportion the House using Hamilton-Vinton, 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 rank-ordered 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 Hamilton-Vinton method is subject to several mathematical anomalies.18

Equal Proportions or Major Fractions: An Analysis

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 (Hill-Huntington) 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.

The Case for Major Fractions

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 value19 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 within-state 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 multi-districted 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.

The Case for Equal Proportions, the Current Method

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 A-1 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.

Appendix. 2010 Priority List for Apportioning Seats to the House of Representatives

Table A-1. 2010 Priority List for Apportioning Seats to the House of Representatives

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.

Footnotes

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 10-year 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. 261-307.

3.

A major controversy occurred even over the fixed, short-term apportionment of seats among the delegates at the Constitutional Convention. See Kromkowski, pp. 287-294.

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, 91st Congress, 2nd session. H. Report 91-1314 (Washington: GPO, 1970), Appendix B, pp. 15-18. Also, see Michel L. Balinski and H. Peyton Young, Fair Representation, 2nd 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 minimum-size 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 435-seat 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, 91st Congress, 2nd session, H. Report 91-1814, (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 85-508 (72 Stat. 345) and section 8 of PL 86-3 (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. 3-4. (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 large-state 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, 97th Congress was introduced by Representative Floyd Fithian and was cosponsored by 10 other members of the Indiana delegation. Changing to the Hamilton-Vinton 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, 97th Cong., 1st sess., June 11, 1981, (Washington: GPO, 1981). Since that time no other bill has been introduced to change the formula.

18.

The Hamilton-Vinton method (used after the 1850-1900 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. Hamilton-Vinton 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 (4-2) is 2. The absolute value of the expression (2-4) 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. CRS84-15), 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: 1995-2025, 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.