The House Apportionment Formula in Theory and Practice

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The House Apportionment Formula

in Theory and Practice

October 10, 2000

David C. Huckabee

Specialist in American National Government

Government and Finance Division

Congressional Research Service ˜ The Library of Congress

The House Apportionment Formula in Theory and Practice

Summary

The Constitution requires that states be represented in the House 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.

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 may 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?

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 unfairness.

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–e.g., Jefferson’s discarded fractions

method, Webster’s major fractions method, the equal proportions method, smallest

divisors method, greatest divisors, the Vinton method, and the Hamilton-Vinton

method. The methodological issues have been problematic for Congress because of

the unfamiliarity and difficulty of some of the mathematical concepts used in the

process.

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 that either the major fractions or the Hamilton-Vinton method be adopted

by Congress as an alternative. A strong 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.

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Constitutional and Statutory Requirements . . . . . . . . . . . . . . . . . . . . . . . . . 2

The Apportionment Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

The Formula In Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Challenges to the Current Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Equal Proportions or Major Fractions: an Analysis . . . . . . . . . . . . . . . . . . 10

The Case for Major Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

The Case for Equal Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Appendix: 1990 Priority List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

List of Tables

Table 1. Multipliers for Determining Priority Values

for Apportioning the House by the Equal Proportions Method . . . . . . . . . . 6

Table 2. Calculating Priority Values for a Hypothetical Three

State House of 30 Seats Using the Method of Equal Proportions . . . . . . . . 6

Table 3. Priority Rankings for Assigning Thirty Seats

in a Hypothetical Three-State House Delegation . . . . . . . . . . . . . . . . . . . . 7

Table 4. Rounding Points for Assigning Seats

Using the Equal Proportions Method of Apportionment* . . . . . . . . . . . . . . 9

The House Apportionment Formula in Theory

and Practice

Introduction

One of the fundamental issues before the framers at the Constitutional

Convention in 1787 was how power was to be allocated in the Congress among the

smaller and larger states. 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 to the states based on the framers’ estimates of how

seats might be apportioned after a census. 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.1

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

(continued...)

CRS-2

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 to 435 after the 1910 Census. 2 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. The congressional debate, which sought an

apportionment method that would minimize those disparities, continued until 1941,

when Congress enacted the “equal proportions” method–the apportionment method

still in use today.

In light of the lengthy debate on apportionment, this report has four major

purposes:

1. to summarize the constitutional and statutory requirements governing

apportionment;

2. to explain how the current apportionment formula works in theory

and in practice;

3. to summarize recent challenges to it on grounds of unfairness; and

4. to explain the reasoning underlying the choice of the equal

proportions method over its chief alternative, 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.3

(...continued)

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 Cong., 2nd sess. H. Rept. 911314 (Washington: GPO, 1970), Appendix B, pp. 15-18.

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 1990 could have been as

small as 50 and as large as 8,301 Representatives.

The actual language in of Article 1, section 2 pertaining to this minimum size reads as

(continued...)

CRS-3

The 1941 apportionment act, in addition to specifying the apportionment

method, sets the House size at 435 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.4

The Census Bureau has been assigned the responsibility of computing the

apportionment. As matter of practice, the Director of the Bureau reports the results

of the apportionment on December 31st of the census year. Once received by

Congress, the Clerk of the House 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.5

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 population6 is divided by the total number of seats in the House (e.g.,

in 1990, 249,022,783 divided by 435) to identify the “ideal” sized congressional

district (572,466 in 1990). 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 which 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

(...continued)

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 mis-read 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.

55 Stat. 761. (1941) Sec. 22 (a). [Codified in 2 U.S.C. 2(a).] In other words, after the 2000

Census, this report is due in January 2001.

Ibid., Sec. 22 (b).

The apportionment population is the population of the 50 states. It excludes the population

of the District of Columbia and U.S. territories and possessions.

CRS-4

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.7 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. In 1990, the

“ideal” size district of 572,466 had to be adjusted upward to between 573,555 and

573,6438 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 twentieth century, Walter F. Willcox, a Cornell University

mathematician, discovered that if the rounding points used in an apportionment

method are divided into each state's population (the mathematical equivalent of

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 in the section analyzing the equal proportions and major

fractions formulas.

Any number in this range divided into each state’s population and rounded at the geometric

mean will produce a 435-seat House.

CRS-5

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.9

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.10

The traditional method of constructing a priority list to apportion seats by the

equal proportions method involves first computing the reciprocals11 of the geometric

means between every pair of consecutive whole numbers (the “rounding points”) so

that it is 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. These reciprocals are

computed for each “rounding point.” They are then used as multipliers to construct

the “priority list.” Table 1 provides a list of multipliers used to calculate the “priority

values” for each state in an equal proportions apportionment.

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, 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 twenty-seven seats must be apportioned using the

equal proportions formula. The 1990 apportionment populations for these states

were 29,839,250 for California, 18,044,505 for New York, and 13,003,362 for

Florida. Table 2 (p. 6) illustrates how the priority values are computed for each state.

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

above, it is possible to see how an increase or decrease in the House size will affect

the allocation of seats without the necessity of doing new calculations. Table 3 (p.

7) ranks the priority values of the three states in this example, showing how the 27

seats are assigned.

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 Cong., 2nd sess., H. Rept. 91-1814, (Washington: GPO, 1970), p. 16.

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. 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.

A reciprocal of a number is that number divided into one.

CRS-6

Table 1. Multipliers for Determining Priority Values

for Apportioning the House by the Equal Proportions Method

Size of

delegation

Multiplier*

Size of

delegation

Multiplier*

Size of

delegation

Multiplier*

Constitution

0.70710678

0.40824829

0.28867513

0.22360680

0.18257419

0.15430335

0.13363062

0.11785113

0.10540926

0.09534626

0.08703883

0.08006408

0.07412493

0.06900656

0.06454972

0.06063391

0.05716620

0.05407381

0.05129892

0.04879500

0.04652421

0.04445542

0.04256283

0.04082483

0.03922323

0.03774257

0.03636965

0.03509312

0.03390318

0.03279129

0.03175003

0.03077287

0.02985407

0.02898855

0.02817181

0.02739983

0.02666904

0.02597622

0.02531848

0.02469324

0.02409813

0.02353104

0.02299002

0.02247333

0.02197935

0.02150662

0.02105380

0.02061965

0.02020305

0.01980295

0.01941839

0.01904848

0.01869241

0.01834940

0.01801875

0.01769981

0.01739196

0.01709464

0.01680732

*Table by CRS, calculated by determining the reciprocals of the geometric means of successive

numbers: 1/ n(n -1) , where “n” is the number of seats to be allocated to the state.

Table 2. Calculating Priority Values for a Hypothetical Three

State House of 30 Seats Using the Method of Equal Proportions

State

Size of

delegation

State’s priority value claim to a delegation size

Calculation

Multiplier (M)

Population (P)

Priority value (PxM)

0.70710678

29,839,250

21,099,536.02

0.40824829

29,839,250

12,181,822.80

0.28867513

29,839,250

8,613,849.51

0.22360680

29,839,250

6,672,259.14

0.18257419

29,839,250

5,447,876.77

0.15430335

29,839,250

4,604,296.24

0.13363062

29,839,250

3,987,437.51

0.11785113

29,839,250

3,516,589.34

0.10540926

29,839,250

3,145,333.12

0.09534626

29,839,250

2,845,060.86

0.08703883

29,839,250

2,597,173.35

0.08006408

29,839,250

2,389,052.01

0.07412493

29,839,250

2,211,832.37

0.06900656

29,839,250

2,059,103.87

0.06454972

29,839,250

1,926,115.31

0.06063391

29,839,250

1,809,270.29

0.05716620

29,839,250

1,705,796.39

0.70710678

18,044,505

12,759,391.85

0.40824829

18,044,505

7,366,638.32

0.28867513

18,044,505

5,208,999.91

0.22360680

18,044,505

4,034,873.98

0.18257419

18,044,505

3,294,460.81

0.15430335

18,044,505

2,784,327.57

CRS-7

Size of

delegation

State

State’s priority value claim to a delegation size

Calculation

Multiplier (M)

Population (P)

Priority value (PxM)

0.13363062

18,044,505

2,411,298.41

0.11785113

18,044,505

2,126,565.31

0.10540926

18,044,505

1,902,057.84

0.09534626

18,044,505

1,720,476.05

0.08703883

18,044,505

1,570,572.57

0.70710678

13,003,362

9,194,765.45

0.40824829

13,003,362

5,308,600.31

0.28867513

13,003,362

3,753,747.28

0.22360680

13,003,362

2,907,640.14

0.18257419

13,003,362

2,374,078.23

0.15430335

13,003,362

2,006,462.32

0.13363062

13,003,362

1,737,647.34

*The “priority values” are the product of the multiplier times the state population. These values can

be computed for any size state delegation, but only those values necessary for this example have been

computed for this table. The population figures are those from the 1990 Census. Table by CRS.

Table 3. Priority Rankings for Assigning Thirty Seats

in a Hypothetical Three-State House Delegation

House

size

State

State’s priority value claim to a delegation size

Size of

Calculation

delegation

Multiplier (M) Population (P) Priority value (PxM)

0.70710678

0.40824829

0.70710678

0.28867513

0.40824829

0.22360680

0.18257419

0.40824829

0.28867513

0.15430335

0.22360680

0.13363062

0.28867513

0.11785113

0.18257419

0.10540926

0.22360680

0.09534626

0.15430335

0.08703883

0.13363062

0.08006408

0.18257419

0.07412493

0.11785113

0.06900656

29,839,250

18,044,505

29,839,250

13,003,362

29,839,250

18,044,505

29,839,250

13,003,362

18,044,505

29,839,250

18,044,505

29,839,250

13,003,362

29,839,250

18,044,505

29,839,250

13,003,362

29,839,250

18,044,505

29,839,250

18,044,505

29,839,250

13,003,362

29,839,250

18,044,505

29,839,250

21,099,536.02

12,759,391.85

12,181,822.80

9,194,765.45

8,613,849.51

7,366,638.32

6,672,259.14

5,447,876.77

5,308,600.31

5,208,999.91

4,604,296.24

4,034,873.98

3,987,437.51

3,753,747.28

3,516,589.34

3,294,460.81

3,145,333.12

2,907,640.14

2,845,060.86

2,784,327.57

2,597,173.35

2,411,298.41

2,389,052.01

2,374,078.23

2,211,832.37

2,126,565.31

2,059,103.87

*The Constitution requires that each state have least one seat. Table by CRS.

CRS-8

From the example in Table 3, 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

15 seats, New York would have nine, and Florida would have six. Any other size

House can be determined by picking points in the priority list and observing what the

maximum size state delegation size would be for each state.

A priority listing for all 50 states based on the 1990 Census is appended 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 4 on the following page 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 which 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 increasingly higher rounding points necessary to obtain additional seats has

led to charges that the equal proportions formula favors small states at the expense

of large states. In a 1982 book about congressional apportionment entitled Fair

Representation, the 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.”12 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 Speaker 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”.13

M.L. Balinski and H.P. Young, Fair Representation, (New Haven and London: Yale

University Press, 1982), p. 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.

“Report of the National Academy of Sciences Committee on Apportionment” in The

Decennial Population Census and Congressional Apportionment, Appendix C, p. 21.

CRS-9

Table 4. Rounding Points for Assigning Seats

Using the Equal Proportions Method of Apportionment*

Size of

delegation

Round

up at

Size of

delegation

1.41421

Round

up at

16.49242

2.44949

4.47214

10.48809

42.49706

57.49783

43.49713

29.49576

58.49786

44.49719

30.49590

56.49779

15.49193

41.49699

28.49561

14.49138

55.49775

40.49691

27.49545

13.49074

54.49771

26.49528

12.49000

53.49766

39.49684

38.49675

25.49510

11.48913

52.49762

37.49667

24.49490

51.49757

23.49468

9.48683

36.49658

22.49444

8.48528

50.49752

35.49648

21.49419

7.48331

49.49747

34.49638

20.49390

6.48074

48.49742

33.49627

19.49359

5.47723

47.49737

46.49731

32.49615

18.49324

Round

up at

Size of

delegation

31.49603

17.49286

3.46410

Round

up at

Size of

delegation

59.49790

45.49725

60.49793

*Any number between 574,847 and 576,049 divided into each state’s 1990 apportionment 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 by CRS.

A bill that would have changed the apportionment method to another formula

called the “Hamilton-Vinton” method was introduced in 1981.14 The fundamental

principle of the Hamilton-Vinton method is that it ranks fractional remainders. To

reapportion the House using Hamilton-Vinton, each state’s population would be

divided by the “ideal” sized congressional district (in 1990, 249,022,783 divided by

435 or 572,466). 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

H.R. 1990 was introduced by Representative Floyd Fithian and was cosponsored by 10

other Members of the Indiana delegation. 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).

CRS-10

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 peculiarities.15

Equal Proportions or Major Fractions: an Analysis

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. The

state of Indiana in 1980 would have been assigned 11 seats under the major fractions

method, and New Mexico would have received 2 seats. Under this allocation, there

would have been 2.004 Representatives per million for Indiana residents and 1.538

Representative per million in New Mexico. The absolute value16 of the difference

between these two numbers is 0.466. Under the equal proportions assignment in

1980, Indiana actually received 10 seats and New Mexico 3. With 10 seats, Indiana

got 1.821 Representatives for each million persons, and New Mexico with 3 seats

received 2.308 Representatives per million. The absolute value of the difference is

0.487. Because major fractions minimizes the absolute population differences, under

it Indiana would have received 11 seats and New Mexico 2, because the absolute

value of subtracting the population shares with an 11 and 2 assignment (0.466) is

smaller than a 10 and 3 assignment (0.487).

An equal proportions apportionment, however, results in a House where the

average sizes of all the states’ congressional districts are as equal as possible if their

differences in size are expressed proportionally–that is, as percentages. The

proportional difference between 2.004 and 1.538 (major fractions) is 30%. The

proportional difference between 2.308 and 1.821 (equal proportions) is 27%. Based

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.

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.

CRS-11

on this comparison, the method of equal proportions gives New Mexico 3 seats and

Indiana 10 because the proportional difference is smaller (27%) than if New Mexico

gets 2 seats and Indiana 10 (30%). From a policy standpoint, one can make a case

for either method by arguing that one measure of fairness is preferable to the other.

The Case for Major Fractions. It can be argued that the major fractions

minimization of absolute size differences among districts most 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).17

Although the “one person, one vote” rules have not been applied by the courts

to apportioning seats among states, major fractions can reduce the range between the

smallest and largest district sizes more than equal proportions–one of the measures

which the courts have applied to within-state redistricting cases. Although this range

would have not changed in 1990, if 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 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).18 Thus, in 1980, shifting from equal proportions to major fractions as a method

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 using simulation techniques.

The House has only been reapportioned 20 times since 1790. The equal

proportions method has been used in five apportionments, and major fractions in

three. Eight apportionments do not provide enough 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 of observing the behavior of systems when

experience does not provide enough information to generalize about them.

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 Court has

never applied its “one person, one vote” rule to apportioning seats–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 for districts

within states unless the size of the House is increased significantly (thereby making districts

smaller).

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).

CRS-12

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. 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.19

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.”20

The Case for Equal Proportions. 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:

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 .03 % of the time. See Fair

Representation, p. 78.

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, p. 13.

CRS-13

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.21

An example using Massachusetts and Oklahoma 1990 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.22 The equal proportions formula assigned 10 seats to Massachusetts

and 6 to Oklahoma in 1990. When 11 seats are assigned to Massachusetts, and five

are given to Oklahoma (using major fractions), Massachusetts has 1.824

Representatives per million persons and Oklahoma has 1.583 Representatives per

million. The absolute difference between these numbers is .241 and the proportional

difference between the two states’ Representatives per million is 15.22%. When 10

seats are assigned to Massachusetts and 6 are assigned to Oklahoma (using equal

proportions), Massachusetts has 1.659 Representatives per million and Oklahoma has

1.9 Representative per million. The absolute difference between these numbers is

.243 and the proportional difference is 14.53%.

Major fractions minimizes absolute differences, so in 1990, if this if this method

had been required by law, Massachusetts and Oklahoma would have received 11 and

five seats respectively because the absolute difference (0.241 Representatives per

million) is smaller at 11 and five than it would be at 10 and 6 (0.243). Equal

proportions minimizes differences on a proportional basis, so it assigned 10 seats to

Massachusetts and six to Oklahoma because the proportional difference between a 10

and 6 allocation (14.53%) is smaller than would occur with an 11 and 5 assignment

(15.22%).

The proportional difference versus absolute difference argument could also be

cast in terms of the goal of “one person, one vote.” 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 different rules govern redistricting in state legislatures than in

Schmeckebier, Congressional Apportionment, p. 60.

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.

CRS-14

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.23

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 which are perhaps appropriate to consider? One such goal might

be the desirability of avoiding geographically large districts, if possible. After the 1990

apportionment, five of the seven states which had only one Representative (Alaska,

Delaware, Montana, North Dakota, South Dakota, Vermont, and Wyoming) have

relatively large land areas.24 The five Representatives of the larger states served

1.27% of the U.S. population, but also represented 27% of the U.S. land area.

Arguably, an apportionment method that would potentially reduce the number

of very large 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 with any certainty, using Census

Bureau projections for 202525 as an illustration, a major fractions apportionment

would result in eight states represented by only one Member, while an equal

proportions apportionment would result in six single-district states.

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.”

The total area of the U.S. is 3,618,770 square miles. The area and (rank) among all states

in area for the seven single district states in this scenario are as follows: Alaska–591,004 (1),

Delaware–2,045 (49), Montana–147,046 (4), North Dakota–70,762 (17), South

Dakota–77,116 (16), Vermont–9,614 (43), Wyoming–97,809 (9). Source: U.S. Department

of Commerce, Bureau of the Census, Statistical Abstract of the United States 1987,

(Washington: GPO, 1987), Table 316: Area of States, p. 181.

U.S. Census Bureau, Projections of the Total Population of States: 1995-2025, Series A,

http://www.census.gov/population/projections/stpjpop.txt, visited Aug. 11, 2000.

CRS-15

The appendix which follows is the priority listing used in reapportionment

following the 1990 Census. This listing 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.

CRS-16

Appendix: 1990 Priority List

Seq. State Seat

Priority

2 21,099,535.65

2 12,759,391.63

3 12,181,821.46

2 12,063,103.59

2 9,194,765.29

4 8,613,849.35

2 8,432,043.16

2 8,108,168.46

2 7,698,501.20

3 7,366,637.51

3 6,964,635.46

5 6,672,258.17

2 6,596,446.31

2 5,479,111.55

6 5,447,875.79

3 5,308,599.72

4 5,208,999.81

4 4,924,741.41

3 4,868,241.93

2 4,707,655.23

3 4,681,252.81

7 4,604,295.11

2 4,602,147.13

3 4,444,731.33

2 4,395,777.31

2 4,263,182.77

5 4,034,873.39

8 3,987,436.09

2 3,934,503.28

5 3,814,687.81

3 3,808,459.70

4 3,753,747.20

2 3,632,975.98

9 3,516,587.79

2 3,469,592.60

2 3,462,447.99

2 3,456,296.16

4 3,442,367.20

2 3,393,138.09

4 3,310,145.91

6 3,294,460.21

3 3,163,366.23

10 3,145,331.61

4 3,142,899.95

6 3,114,679.44

2 3,102,097.90

2 2,996,871.22

5 2,907,639.71

2 2,872,697.61

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

2,845,059.46

2,784,326.89

2,717,965.76

2,692,987.92

2,666,445.82

2,657,050.63

2,632,384.41

2,615,566.01

2,600,728.09

2,597,172.96

2,564,027.67

2,537,902.98

2,478,909.15

2,461,349.49

2,434,479.52

2,411,297.55

2,389,051.45

2,374,077.80

2,339,046.96

2,330,389.85

2,279,711.53

2,271,586.31

2,236,837.92

2,232,763.16

2,211,830.60

2,177,143.82

2,126,564.37

2,097,499.46

2,093,519.75

2,085,979.21

2,059,102.28

2,017,893.92

2,010,516.41

2,006,461.82

2,003,170.03

1,999,045.09

1,995,493.33

1,987,744.13

1,971,006.37

1,959,029.01

1,926,114.17

1,921,892.20

1,902,056.92

1,878,818.69

1,840,022.25

1,828,891.35

1,809,270.25

1,798,260.48

1,794,568.57

1,790,996.89

CRS-17

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

1,769,347.01

1,757,584.58

1,740,437.07

1,737,646.72

1,732,646.98

1,730,244.24

1,720,475.20

1,705,796.31

1,703,194.83

1,679,950.30

1,670,355.18

1,658,552.58

1,626,587.79

1,613,521.84

1,606,254.23

1,593,505.83

1,570,572.33

1,532,460.23

1,532,299.29

1,530,721.18

1,510,097.60

1,501,530.92

1,488,691.10

1,484,865.21

1,483,156.23

1,456,006.30

1,455,326.51

1,454,879.48

1,444,716.30

1,439,462.27

1,431,198.73

1,416,455.24

1,414,700.28

1,413,538.47

1,411,027.00

1,405,339.93

1,390,066.66

1,388,247.47

1,385,242.82

1,370,674.05

1,365,877.22

1,351,360.84

1,350,449.27

1,348,136.59

1,345,451.08

1,337,546.63

1,326,516.39

1,289,086.29

1,283,082.99

1,273,941.23

1,270,042.73

1,266,426.16

1,264,555.87

1,256,974.20

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

1,246,610.75

1,245,188.18

1,244,199.02

1,239,821.31

1,223,467.55

1,221,727.76

1,218,182.21

1,215,511.15

1,208,693.83

1,195,639.89

1,188,269.08

1,177,237.47

1,172,773.89

1,170,391.58

1,165,031.49

1,164,767.10

1,148,847.73

1,147,624.27

1,137,960.95

1,136,975.93

1,134,984.63

1,131,797.21

1,126,209.87

1,120,493.40

1,101,205.03

1,100,748.87

1,099,407.25

1,097,181.37

1,094,922.05

1,094,108.80

1,093,304.69

1,092,976.67

1,085,243.01

1,076,060.23

1,073,004.34

1,067,800.35

1,061,742.79

1,055,910.81

1,047,152.30

1,041,101.93

1,038,065.20

1,037,912.62

1,035,454.40

1,034,402.59

1,031,535.64

1,027,294.36

1,015,884.21

1,014,741.83

1,012,010.42

1,011,645.28

1,004,270.60

998,046.41

983,339.70

980,969.36

CRS-18

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

978,467.51

975,735.07

975,244.09

964,379.92

963,872.55

959,237.03

954,911.92

954,740.67

951,377.67

947,693.77

947,619.86

947,397.10

938,030.21

930,302.53

925,663.55

922,488.60

918,239.42

918,069.09

913,184.87

911,521.74

908,426.63

897,316.53

895,844.81

894,000.08

892,411.68

890,823.07

889,662.91

889,464.22

883,917.61

880,481.68

876,104.34

875,149.50

872,020.33

871,683.42

869,723.76

864,996.63

858,578.81

852,878.24

849,966.34

840,625.60

839,506.30

839,362.91

832,433.24

830,723.54

827,114.49

823,801.74

822,882.53

822,422.32

817,590.39

816,777.34

811,966.30

807,021.57

805,665.54

804,659.98

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

364

365

802,176.05

800,958.10

795,784.05

793,693.91

792,780.17

791,275.62

788,617.79

788,444.61

787,656.83

784,608.87

783,899.80

775,110.76

773,788.69

769,736.26

768,025.08

767,024.19

758,400.80

757,127.00

755,567.92

755,484.48

754,225.48

751,296.22

746,900.30

746,641.76

743,550.98

743,352.69

741,727.21

740,443.26

740,170.70

739,671.50

738,802.90

736,933.88

736,827.74

736,663.79

735,510.24

732,629.24

726,113.47

723,041.73

719,070.17

717,530.90

715,582.15

711,338.09

710,530.16

707,763.66

706,061.61

705,364.78

703,141.28

702,773.39

702,148.53

701,775.48

696,463.58

695,269.70

691,494.92

686,567.69

CRS-19

366

367

368

369

370

371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

686,047.27

686,005.00

681,919.64

681,690.26

681,045.92

676,933.10

675,336.13

674,431.92

671,504.99

670,587.24

669,140.55

667,058.37

660,141.03

656,272.29

655,847.22

655,750.27

655,691.14

655,506.55

655,283.49

654,340.94

653,970.76

653,178.36

646,917.11

644,814.46

643,880.82

643,746.75

641,741.37

641,242.61

640,051.48

638,114.00

635,517.47

634,779.80

634,499.09

633,237.93

628,229.44

626,873.87

623,286.86

622,386.91

621,263.60

620,553.10

620,459.09

620,387.08

620,047.14

615,274.87

611,765.99

611,724.70

605,495.74

604,971.11

603,939.23

602,843.86

602,170.06

601,703.97

598,681.74

592,726.21

420 NY

591,702.60

421 CA

590,905.18

422 OH

588,719.10

423

588,228.36

424

586,520.84

425 MN

586,241.20

426 PA

581,866.26

427 NC

579,472.22

428 CA

579,430.15

429 TX

578,381.53

430 MS

578,346.15

431 WS

578,265.19

432

578,069.92

433 TN

577,074.42

434 OK

576,496.87

435 WA

576,049.11

Last seat assigned by law

436 MA

574,847.17

437 NJ

574,366.50

438 NY

572,913.58

439 KY

570,763.16

440 CA

568,392.42

441 MT

568,269.89

442 AZ

567,525.26

443 GA

566,485.07

444 LA

566,355.23

445 MG 17

565,640.60

446 MD

565,522.77

447

559,516.78

448 TX

559,413.02

449 OH

558,507.97

450 CA

557,767.31

451 KA

555,796.97

452 NY

555,281.24

453 PA

554,787.68

454

553,459.80

455 CA

547,532.16

456 AL

542,888.63

457 TX

541,649.33

458 MO 10

541,571.83

459 VA

541,082.71

460 SC

540,942.20

461 NY

538,701.92

462 CA

537,665.94

463 NJ

534,706.13

464

533,478.29

465 MG 18

533,291.06

466 NC

533,036.87

467 OH

531,247.06

468

530,860.00

469

530,528.06

470 PA

530,117.99

471 AR

528,212.62

472 CA

528,148.99

CRS-20

473

474

475

476

477

478

479

480

481

482

483

484

485

486

487

488

489

490

491

492

493

494

495

496

497

498

499

500

524,979.20

524,761.45

523,084.05

521,090.43

521,017.88

520,084.33

518,963.07

517,216.08

517,016.09

516,151.03

515,233.97

510,421.77

510,091.18

510,033.77

509,756.16

509,304.61

508,911.57

508,532.64

508,346.32

507,549.32

506,524.17

505,818.92

504,442.86

503,461.12

501,517.64

500,171.88

499,478.32

498,768.26

This is a copy of a public record, reproduced as it was published. It is not legal advice, and it may not be the version a court would rely on. Check the official source before you cite it.