October 10, 2000

Royce Crocker

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

In troduction ..................................................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

1

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.

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.

(continued...)

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.

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.

Subsequently, mathematician W.F. Willcox proposed the “major fractions” method, which

was used following the census of 1910. This method, too, had its critics; and in 1921

Harvard mathematician E.V. Huntington proposed the “equal proportions” method and

developed formulas and computational tables for all of the other known, mathematically

valid apportionment methods. A committee of the National Academy of Sciences conducted

an analysis of each of those methods — smallest divisors, harmonic mean, equal

proportions, major fractions, and greatest divisors — and recommended that Congress adopt

Huntington’s equal proportions method. For a review of this history, see U.S. Congress,

House, Committee on Post Office and Civil Service, Subcommittee on Census and Statistics,

The Decennial Population Census and Congressional Apportionment, 91 Cong., 2 sess.

H. Rept. 91-1314 (Washington: GPO, 1970), Appendix B, pp. 15-18.

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.

(continued...)

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.

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.

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 population

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

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.

Census, this report is due in January 2001.

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

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.

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 573,643

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

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.

mean will produce a 435-seat House.

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.

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.

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

equal proportions method involves first computing the reciprocals

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.

Census and Statistics, The Decennial Population Census and Congressional Apportionment,

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

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.

Table 1. Multipliers for Determining Priority Values

for Apportioning the House by the Equal Proportions Method

Table 2. Calculating Priority Values for a Hypothetical Three

State House of 30 Seats Using the Method of Equal Proportions

Table 3. Priority Rankings for Assigning Thirty Seats

in a Hypothetical Three-State House Delegation

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.”

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”.

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.

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

Table 4. Rounding Points for Assigning Seats

Using the Equal Proportions Method of Apportionment*

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

called the “Hamilton-Vinton” method was introduced in 1981. 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

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, 97 Cong., 1 sess., June 11, 1981, (Washington: GPO, 1981).

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.

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

value

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

“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 -8 is 8. The absolute value of the expression (4-2) is 2. The absolute

value of the expression (2-4) is also 2.

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).^{18}

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).

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.

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).

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).

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.

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.”

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:

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

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.

a contract for the Congressional Research Service of the Library of Congress. (Contract No.

CRS84-15), Sept. 30, 1984, p. 13.

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.

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.

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

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

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.

from redistricting within states — proportional difference measures might be

accepted as most appropriate.

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. 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 2025 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.

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.”

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.

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

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.

Appendix: 1990 Priority List