August 10, 2001

Royce Crocker

Specialist in American National Government

Government and Finance Division

Congressional Research Service ˜ The Library of Congress

The House of Representatives Apportionment Formula:

An Analysis of Proposals for Change

and Their Impact on States

Summary

Now that the reallocation of Representatives among the states based on the 2000

Census has been completed, some members of the statistical community are urging

Congress to consider changing the current House apportionment formula. However,

other formulas also raise questions.

Seats in the House of Representatives are allocated by a formula known as the

Hill, or equal proportions, method. If Congress decided to change it, there are at least

five alternatives to consider. Four of these are based on rounding fractions; one, on

ranking fractions. The current apportionment system (codified in 2 U.S.C. 2a) is one

of the rounding methods.

The Hamilton-Vinton method is based on ranking fractions. First, the

population of 50 states is divided by 435 (the House size) in order to find the national

“ideal size” district. Next this number is divided into each state’s population. Each

state is then awarded the whole number in its quotient (but at least one). If fewer

than 435 seats have been assigned by this process, the fractional remainders of the

50 states are rank-ordered from largest to smallest, and seats are assigned in this

manner until 435 are allocated.

The rounding methods, including the Hill method currently in use, allocate seats

among the states differently, but operationally the methods only differ by where

rounding occurs in seat assignments. Three of these methods — Adams, Webster,

and Jefferson — have fixed rounding points. Two others — Dean and Hill — use

varying rounding points that rise as the number of seats assigned to a state grows

larger. The methods can be defined in the same way (after substituting the

appropriate rounding principle in parentheses). The rounding point for Adams is (up

for all fractions); for Dean (at the harmonic mean); for Hill (at the geometric mean);

for Webster (at the arithmetic mean — .5); and for Jefferson (down for all fractions).

Substitute these phrases in the general definition below for the rounding methods:

Find a number so that when it is divided into each state’s population and

resulting quotients are rounded (substitute appropriate phrase), the total

number of seats will sum to 435. (In all cases where a state would be

entitled to less than one seat, it receives one anyway because of the

constitutional requirement.)

Unlike the Hamilton-Vinton method, which uses the national “ideal size”

district for a divisor, the rounding methods use a sliding divisor. If the national

“ideal size” district results in a 435-seat House after rounding according to the rule

of method, no alteration in its size is necessary. If too many seats are allocated, the

divisor is made larger (it slides up); if too few seats are apportioned, the divisor

becomes smaller (it slides down). Fundamental to choosing an apportionment

method is a determination of fairness. Each of the competing formulas is the best

method for satisfying one or more mathematical tests.

Contents

In troduction ......................................................1

Background ......................................................3

Apportionment Methods Defined.....................................5

Hamilton-Vinton: Ranking Fractional Remainders...................5

Rounding Methods.............................................8

Webster: Rounding at the Midpoint (.5).......................10

Hill: Rounding at the Geometric Mean........................10

Dean: Rounding at the Harmonic Mean.......................11

Adams: All Fractions Rounded Up...........................11

Jefferson: All Fractions Rounded Down.......................12

Changing the Formula: The Impact in 2001............................12

A Framework for Evaluating Apportionment Methods....................17

Alternative Kinds of Tests......................................19

Fairness and Quota............................................21

Quota Representation......................................21

Implementing the “Great Compromise”...........................22

Conclusion ......................................................23

List of Figures

Figure 1. Illustrative Rounding Points for Five Apportionment

Methods (for Two and Twenty-one Seats)..........................9

List of Tables

Table 1. Apportioning the House in 2001 by Simple Rounding

and Ranked Fractional Remainders (Hamilton-Vinton)................7

Table 2. Seat Assignments in 2001 for Various

House Apportionment Formulas (Alphabetical Order)................13

Table 3. Seat Assignments in 2001 for Various

House Apportionment Formulas (Ranked by State Population).........15

Table 4. Alternate Methods for Measuring Equality

of District Sizes..............................................20

1

Introduction

Now that the reallocation of Representatives among the states based on the 2000

Census has been completed, some members of the statistical community are urging

Congress to consider changing the current House apportionment formula. However,

other formulas also raise questions.

In 1991, the reapportionment of the House of Representatives was nearly

overturned because the current “equal proportions” formula for the House

apportionment was held to be unconstitutional by a three-judge federal district court.

The court concluded that:

By complacently relying, for over fifty years, on an apportionment method which

does not even consider absolute population variances between districts, Congress

has ignored the goal of equal representation for equal numbers of people. The

court finds that unjustified and avoidable population differences between

districts exist under the present apportionment, and ... [declares] section 2a of

Title 2, United States Code unconstitutional and void.

The three-judge panel’s decision came almost on the 50

formula’s enactment.

The government appealed the panel’s decision to the Supreme Court, where

Montana argued that the equal proportions formula violated the Constitution because

it “does not achieve the greatest possible equality in number of individuals per

Representative.” This reasoning did not prevail, because, as Justice Stevens wrote

in his opinion for a unanimous court, absolute and relative differences in district sizes

Reinstate the Old Reapportionment Formula, by H. Peyton Young, Policy Brief No. 88

(Washington, Brookings Institution, August 2001). Young suggests that Congress consider

the matter “now — well in advance of the next census,” p. 1.

District Court for the District of Montana, Helena Division.

are identical when considering deviations in district populations within states, but

they are different when comparing district populations among states. Justice Stevens

noted, however, that “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 year 1991 was a banner year for court challenges on the apportionment

front. At the same time the Montana case was being argued, another case was being

litigated by Massachusetts. The Bay State lost a seat to Washington because of the

inclusion of 978,819 federal employees stationed overseas in the state populations

used to determine reapportionment. The court ruled that Massachusetts could not

challenge the President’s decision to include the overseas federal employees in the

apportionment counts, in part because the President is not subject to the terms of the

Administrative Procedures Act.

In 2001, the Census Bureau’s decision to again include the overseas federal

employees in the population used to reapportion the House produced a new challenge

to the apportionment population. Utah argued that it lost a congressional seat to

North Carolina because of the Bureau’s decision to include overseas federal

employees in the apportionment count, but not other citizens living abroad. Utah

said that Mormon missionaries were absent from the state because they were on

assignment: a status similar to federal employees stationed overseas. Thus, the state

argued, the Census Bureau should have included the missionaries in Utah’s

apportionment count. The state further argued that, unlike other U.S. citizens living

overseas, missionaries could be accurately reallocated to their home states because

the Mormon church has excellent administrative records. Utah’s complaint was

dismissed by a three-judge federal court on April 17, 2001.

The Supreme Court appears to have settled the issue about Congress’s discretion

to choose a method to apportion the House, and has granted broad discretion to the

President in determining who should be included in the population used to allocate

seats. Although modern Congresses have rarely considered the issue of the formula

(APA) sets forth the procedures by which federal agencies are accountable to the public and

their actions are subject to review by the courts. Since the Supreme Court ruled that a

President’s decisions are not subject to review under the APA by courts, the district court’s

decision to the contrary was reversed. Plaintiffs in this case also challenged the House

apportionment formula, arguing that the Hill (equal proportions) method discriminated

against larger states.

Representative Gilman introduced H.R. 1745, the Full Equality for Americans Abroad Act,

on May 8, 2001. The bill would require including all citizens living abroad in the state

populations used for future apportionments. For further reading on this and other legal

matters pertaining to the 2000 census, see CRS Report RL30870, Census 2000: Legal Issues

re: Data for Reapportionment and Redistricting, by Margaret Mikyung Lee.

used in the calculations, this report describes apportionment options from which

Congress could choose and the criteria that each method satisfies.

Background

One of the fundamental issues before the framers at the constitutional

convention in 1787 was how power was to be allocated in Congress between the

smaller and larger states. The solution ultimately adopted became known as the

Great (or Connecticut) Compromise. It solved the controversy between large and

small states by creating a bicameral Congress with states equally represented in the

Senate and seats allocated by population in the House. The Constitution provided the

first apportionment: 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

clause 2 of the Fourteenth Amendment:

Amendment XIV, section 2. Representatives and direct taxes 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 ....

The constitutional mandate that Representatives would be apportioned

according to population did not describe how Congress was to distribute fractional

entitlements to Representatives. Clearly there would be fractions because districts

could not cross state lines and the states’ populations were unlikely to be evenly

divisible. From its beginning in 1789 Congress was faced with deciding how to

apportion the House of Representatives. The controversy continued until 1941, with

the enactment of the Hill (“equal proportions”) method. During congressional

debates on apportionment, the major issues were how populous a congressional

district ought to be (later re-cast as how large the House ought to be), and how

fractional entitlements to Representatives should be treated. The matter of the

permanent House size has received little attention since it was last increased to 435

after the 1910 Census.

picture — determining which method comes closest to meeting the goal of “one

person, one vote.”

Congress to adopt the Hamilton-Vinton method of apportionment to be effective for the

1980 and subsequent censuses. Hearings were held in the House, but no further action was

taken.

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 2001 could have been as

small as 50 and as large as 9,361 Representatives (30,000 divided into the total U.S.

apportionment population).

The “one person, one vote” concept was established through a series of Supreme

Court decisions beginning in the 1960s. The court ruled in 1962 that state legislative

districts must be approximately equal in population (Baker v. Carr, 369 U.S. 186).

This ruling was extended to the U.S. House of Representatives in 1964 (Wessberry

v. Sanders, 376 U.S. 1). Thus far, the concept has only been applied within states.

states must be able to justify any deviations from absolute numerical equality for

their congressional districts in order to comply with a 1983 Supreme Court decision

— Karcher v. Daggett (462 U.S. 725).

The population distribution among states in the 2000 Census, combined with a

House size of 435, and the requirement that districts not cross state lines, means that

there is a wide disparity in district sizes — from 495,304 (Wyoming) to 905,316

(Montana) after the 2000 Census. This interstate population disparity among

districts in 2001 contrasts with the intrastate variation experienced in the

redistrictings following the 1990 Census. Nineteen of the 43 states that had two or

more districts in 1992 drew districts with a population difference between their

districts of ten persons or fewer, and only six states varied by more than 1,000

persons.

Given a fixed-size House and an increasing population, there will inevitably be

population deviations in district sizes among states; what should be the goal of an

apportionment method? Although Daniel Webster was a proponent of a particular

formula (the major fractions method), he succinctly defined the apportionment

problem during debate on an apportionment bill in 1832. Webster said that:

The Constitution, therefore, must be understood, not as enjoining an absolute

relative equality, because that would be demanding an impossibility, but as

requiring of Congress to make the apportionment of Representatives among the

several states according to their respective numbers, as near as may be. That

which cannot be done perfectly must be done in a manner as near perfection as

can be ....

Which apportionment method is the “manner as near perfection as can be”?

Although there are potentially thousands of different ways in which the House can

be apportioned, six methods are most often mentioned as possibilities. These are the

methods of: Hamilton-Vinton, “largest fractional remainders”; Adams, “smallest

divisors”; Dean, “harmonic mean”; Hill, “equal proportions”; Webster, “major

fractions”; and Jefferson, “largest divisors.”

Process, by Royce Crocker, pp. 53-54.

Institution Press, 2001), p. 31.

Apportionment Methods Defined

Hamilton-Vinton: Ranking Fractional Remainders

Why is there a controversy? Why not apportion the House the intuitive way by

dividing each state’s population by the national “ideal size” district (645,632 in 2001)

and give each state its “quota” (rounding up at fractional remainders of .5 and above,

and down for remainders less than .5)? The problem with this proposal is that the

House size would fluctuate around 435 seats. In some decades, the House might

include 435 seats; in others, it might be either under or over the legal limit. In 2001,

this method would result in a 433-seat House (438 in 1991).

One solution to this problem of too few or too many seats would be to divide

each state’s population by the national “ideal” size district, but instead of rounding

at the .5 point, allot each state initially the whole number of seats in its quota (except

that states entitled to less than one seat would receive one regardless). Next, rank the

fractional remainders of the quotas in order from largest to smallest. Finally, assign

seats in rank order until 435 are allocated (see Table 1). If this system had been used

in 2001, California would have one less Representative, and Utah would have one

more.

This apportionment formula, which is associated with Alexander Hamilton, was

used in Congress’s first effort to enact an apportionment of the House. The bill was

vetoed by President Washington — his first exercise of this power.

which might be described as the largest fractional remainders method, was used by

Congress from 1851 to 1901;

were made in the apportionments that were not consistent with the method.

generally been known as the Vinton method (for Representative Samuel Vinton

(Ohio), its chief proponent after the 1850 Census). Assuming a fixed House size, the

Hamilton-Vinton method can be described as follows:

Hamilton-Vinton

Divide the apportionment population

the “ideal congressional district size” to be used as a divisor. Divide each

state’s population by the ideal size district to obtain its quota. Award each

state the whole number obtained in these quotas. (If a state receives less

than one Representative, it automatically receives one because of the

constitutional requirement.) If the number of Representatives assigned

using the whole numbers is less than the House total, rank the fractional

Institution, 1941). p. 73.

remainders of the states’ quotas and award seats in rank order from

highest to lowest until the House size is reached.

The Hamilton-Vinton method has simplicity in its favor, but its downfall was

the Alabama paradox. Although the phenomenon had been observed previously, the

“paradox” became an issue after the 1880 census when C. W. Seaton, Chief Clerk of

the Census Office, wrote the Congress on October 25, 1881, stating:

While making these calculations I met with the so-called “Alabama” paradox

where Alabama was allotted 8 Representatives out of a total of 299, receiving but

7 when the total became 300.

Alabama’s loss of its eighth seat when the House size was increased resulted

from the vagaries of fractional remainders. With 299 seats, Alabama’s quota was

7.646 seats. It was allocated eight seats based on this quota, but it was on the

dividing point. When a House size of 300 was used, Alabama’s quota increased to

7.671, but Illinois and Texas now had larger fractional remainders than Alabama.

Accordingly, each received an additional seat in the allotment of fractional

remainders, but since the House had increased in size by only one seat, Alabama lost

the seat it had received in the allotment by fractional remainders for 299 seats.

property of the Hamilton-Vinton method became a big enough issue that the formula

was changed in 1911.

One could argue that the Alabama paradox should not be an important

consideration in apportionments, since the House size was fixed in size at 435, but

the Hamilton-Vinton method is subject to other anomalies. Hamilton-Vinton is also

subject to the population paradox and the new states paradox.

The population paradox occurs when a state that grows at a greater percentage

rate than another has to give up a seat to the slower growing state. The new states

paradox works in much the same way — at the next apportionment after a new state

enters the Union, any increase in House size caused by the additional seats for the

new state may result in seat shifts among states that otherwise would not have

happened. Finding a formula that avoided the paradoxes was a goal when Congress

adopted a rounding, rather than a ranking, method when the apportionment law was

changed in 1911.

Table 1 illustrates how a Hamilton-Vinton apportionment would be done by

ranking the fractional remainders of the state’s quotas in order from largest to

smallest. In 2001 North Carolina and Utah’s fractional remainders of less than 0.5

would have been rounded up by the Hamilton-Vinton method in order for the House

to have totaled 435 Representatives.

Table 1. Apportioning the House in 2001 by Simple Rounding

and Ranked Fractional Remainders (Hamilton-Vinton)

Rounding Methods

The kinds of calculations required by the Hamilton-Vinton method are

paralleled, in their essentials, in all the alternative methods that are most frequently

discussed — but fractional remainders are rounded instead of ranked. First, the total

apportionment population, (the population of the 50 states as found by the census)

is divided by 435, or the size of the House. This calculation yields the national

“ideal” district size. Second, the “ideal” district size is used as a common divisor for

the population of each state, yielding what are called the states’ quotas of

Representatives. Because the quotas still contain fractional remainders, each method

then obtains its final apportionment by rounding its allotments either up or down to

the nearest whole number according to certain rules.

The operational difference between the methods lies in how each defines the

rounding point for the fractional remainders in the allotments — that is, the point at

which the fractions rounded down are separated from those rounded up. Each of the

rounding methods defines its rounding point in terms of some mathematical quantity.

Above this specified figure, all fractional remainders are automatically rounded up;

those below, are rounded down.

For a given common divisor, therefore, each rounding method yields a set

number of seats. If using national “ideal” district size as the common divisor results

in 435 seats being allocated, no further adjustment of the divisor is necessary. But

if too many or too few seats are apportioned, the common divisor must be varied

until a value is found that yields the desired number of seats. (These methods will,

as a result, generate allocations before rounding that differ from the states’ quotas.)

If too many seats are apportioned, a larger divisor is tried (the divisor slides up); if

too few, a smaller divisor (it slides down). The divisor finally used is that which

apportions of a number of seats equal to the desired size of the House.

they use a common divisor. This report characterizes them as rounding methods, although

they use common divisors, because the Hamilton-Vinton method also uses a common

divisor, while its actual apportionment is not based on rounding. All these methods can be

described in different ways, but looking at them based on how they treat quotients provides

(continued...)

Figure 1. Illustrative Rounding Points for Five Apportionment

Methods (for Two and Twenty-one Seats)

The rounding methods that are mentioned most often (although there could be

many more) are the methods of: Webster (“major fractions”); Hill (“equal

proportions” — the current method); Dean (“harmonic mean”); Adams (“smallest

divisors”); and Jefferson (“greatest divisors”). Under any of these methods, the

Census Bureau would construct a priority list of claims to representation in the

House.

rounding point is set — that is, the rule that determines what fractional remainders

result in a state being rounded up, rather than down.

In the Adams, Webster, and Jefferson methods, the rounding points used are the

same for a state of any size. In the Dean and Hill methods, on the other hand, the

rounding point varies with the number of seats assigned to the state; it rises as the

the state’s population increases. With these two methods, in other words, smaller

(less populous) states will have their apportionments rounded up to yield an extra

seat for smaller fractional remainders than will larger states. This property provides

a consistent framework to understand them all.

Report RL30711, The House Apportionment Formula in Theory and Practice, by Royce

Crocker.

the intuitive basis for challenging the Dean and Hill methods as favoring small states

at the expense of the large (more populous) states.

These differences among the rounding methods are illustrated in Figure 1. The

“flags” in Figure 1 indicate the points that a state’s fractional remainder must exceed

for it to receive a second seat, and to receive a 21st seat. Figure 1 visually illustrates

that the only rounding points which change their relative positions are those for Dean

and Hill. Using the rounding points for a second seat as the example, the Adams

method awards a second seat for any fractional remainder above one. Dean awards

the second seat for any fractional remainder above 1.33. Similarly, Hill gives a

second seat for every fraction exceeding 1.41, Webster, 1.5, and Jefferson does not

give a second seat until its integer value of a state’s quotient equals or exceeds two.

Webster: Rounding at the Midpoint (.5). The easiest rounding method

to describe is the Webster (“major fractions”) method which allocates seats by

rounding up to the next seat when a state has a remainder of .5 and above. In other

words, it rounds fractions to the lower or next higher whole number at the arithmetic

mean, which is the midpoint between numbers. For example, between 1 and 2 the

arithmetic mean is 1.5; between 2 and 3, the arithmetic mean is 2.5, etc. The

Webster method (which was used in 1840, 1910, and 1930) can be defined in the

following manner for a 435-seat House:

Webster

Find a number so that when it is divided into each state’s population and

resulting quotients are rounded at the arithmetic mean, the total number

of seats will sum to 435. (In all cases where a state would be entitled to

less than one seat, it receives one anyway because of the constitutional

entitlement.)

Hill: Rounding at the Geometric Mean. The only operational difference

between a Webster and a Hill apportionment (equal proportions — the method in use

since 1941), is where the rounding occurs. Rather than rounding at the arithmetic

mean between the next lower and the next higher whole number, Hill rounds at the

geometric mean. The geometric mean is the square root of the multiplication of two

numbers. The Hill rounding point between 1 and 2, for example, is 1.41 (the square

root of 2), rather than 1.5. The rounding point between 10 and 11 is the square root

of 110, or 10.487. The Hill method can be defined in the following manner for a

435-seat House:

percent.” He determined this figure by first eliminating from the calculations the very small

states whose quotas equaled less than one half a Representative. He then computed the

relative bias for the methods described in this report for all the censuses based on the “per

capita representation in the large states as a group and in the small states as group. The

percentage difference between the two is the method’s relative bias toward small states in

that year. To estimate their long-run behavior, I compute the average bias of each method

up to that point in time.” See: Brookings Institution Policy Brief No. 88, Dividing the

House: Why Congress Should Reinstate the Old Reapportionment Formula, p. 4.

Hill

Find a number so that when it is divided into each state’s population and

resulting quotients are rounded at the geometric mean, the total number

of seats will sum to 435. (In all cases where a state would be entitled to

less than one seat, it receives one anyway because of the constitutional

entitlement.)

Dean: Rounding at the Harmonic Mean. The Dean method (advocated

by Montana) rounds at a different point — the harmonic mean between consecutive

numbers. The harmonic mean is obtained by multiplying the product of two numbers

by 2, and then dividing that product by the sum of the two numbers.

rounding point between 1 and 2, for example, is 1.33, rather than 1.5. The rounding

point between 10 and 11 is 10.476. The Dean method (which has never been used)

can be defined in the following manner for a 435-seat House:

Dean

Find a number so that when it is divided into each state’s population and

resulting quotients are rounded at the harmonic mean, the total number of

seats will sum to 435. (In all cases where a state would be entitled to less

than one seat, it receives one anyway because of the constitutional

entitlement.)

Adams: All Fractions Rounded Up. The Adams method (“smallest

divisors”) rounds up to the next seat for any fractional remainder. The rounding point

between 1 and 2, for example, would be any fraction exceeding 1 with similar

rounding points for all other integers. The Adams method (which has never been

used, but is also advocated by Montana) can be defined in the following manner for

a 435-seat House:

Adams

Find a number so that when it is divided into each state’s population and

resulting quotients that include fractions are rounded up, the total number

of seats will sum to 435. (In all cases where a state would be entitled to

less than one seat, it receives one anyway because of the constitutional

entitlement.)

2*(A*B)/(A+B).

Jefferson: All Fractions Rounded Down. The Jefferson method (“largest

divisors”) rounds down any fractional remainder. In order to receive 2 seats, for

example, a state would need 2 in its quotient, but it would not get 3 seats until it had

3 in its quotient. The Jefferson method (used from 1790 to 1830) can be defined in

the following manner for a 435-seat House:

Jefferson

Find a number so that when it is divided into each state’s population and

resulting quotients that include fractions are rounded down, the total

number of seats will sum to 435. (In all cases where a state would be

entitled to less than one seat, it receives one anyway because of the

constitutional requirement.)

Changing the Formula: The Impact in 2001

What would happen in 2001 if any of the alternative formulas discussed in this

report were to be adopted? As compared to the Hill (equal proportions)

apportionment currently mandated by law, the Dean method, advocated by Montana

in 1991, results (not surprisingly) in Montana regaining its second seat that it lost in

1991, and Utah gaining a fourth seat. Neither California nor North Carolina would

have gained seats in 2001 using the Dean method. The Webster method would have

caused no change in 2001, but in 1991 it would have resulted in Massachusetts

retaining a seat it would otherwise would have lost under Hill, while Oklahoma

would have lost a seat. The Hamilton-Vinton method (as discussed earlier) results

in Utah gaining and California not gaining a seat as compared to the current (Hill)

method. The Adams method in 2001 would reassign eight seats among fourteen

states (see Table 2). The Jefferson method would reassign six seats among twelve

states (see Table 2).

Tables 2 and 3, which follow, present seat allocations based on the 2000

Census for the six methods discussed in this report. Table 2 is arranged in

alphabetical order. Table 3 is arranged by total state population, rank-ordered from

the most populous state (California) to the least (Wyoming). This table facilitates

evaluating apportionment methods by looking at their impact according to the size

of the states. Allocations that differ from the current method are bolded and italicized

in both tables.

Table 2. Seat Assignments in 2001 for Various House

Apportionment Formulas (Alphabetical Order)

Table 3. Seat Assignments in 2001 for Various House

Apportionment Formulas

(Ranked by State Population)

A Framework for Evaluating

Apportionment Methods

All the apportionment methods described above arguably have properties that

recommend them. Each is the best formula to satisfy certain mathematical measures

of fairness, and the proponents of some of them argue that their favorite meets other

goals as well. The major issue raised in the Montana case

approximates the “one person, one vote” principle. The apportionment concerns

raised in the Massachusetts case

also suggested that the Hill method discriminates against the larger states.

It is not immediately apparent which of the methods described above is the

“fairest” or “most equitable” in the sense of meeting the “one person, one vote”

standard. As already noted, no apportionment formula can equalize districts

precisely, given the constraints of (1) a fixed size House, (2) a minimum seat

allocation of one, and (3) the requirement that districts not cross state lines. The

practical question to be answered, therefore, is not how inequality can be eliminated,

but how it can be minimized. This question too, however, has no clearly definitive

answer, for there is no single established criterion by which to determine the equality

or fairness of a method of apportionment.

In a report to the Congress in 1929, the National Academy of Sciences (NAS)

defined a series of possible criteria for comparing how well various apportionment

formulas achieve equity among states.

enunciation of the “one person, one vote” principle by more than 30 years, but if the

Congress decided to reevaluate its 1941 choice to adopt the Hill method, it could use

one of the NAS criteria of equity as a measure of how well an apportionment formula

fulfills that principle.

Although the following are somewhat simplified restatements of the NAS

criteria, they succinctly present the question before the Congress if it chose to take

up this matter. Which of these measures best approximates the one person, one vote

concept?

average district size in the country and the smallest? This criterion

leads to the Dean method.

individual share of his or her Representative by subtracting the

Census and Statistics, The Decennial Population Census and Congressional

Apportionment, Appendix C: Report of National Academy of Sciences Committee on

Apportionment, 91 Cong., 1 Sess., H.Rept. 91-1314 (Washington: GPO, 1970), pp. 19-21.

largest such share for a state from the smallest share? This criterion

leads to the Webster method.

or in individual shares of a Representative, when those differences

are expressed as percentages? These criteria both lead to the Hill

method.

among states?

among states?

In the absence of further information, it is not apparent which criterion (if any)

best encompasses the principle of “one person, one vote.” Although the NAS report

endorsed as its preferred method of apportionment the one currently in use — the

Hill method — the report arguably does not make a clear-cut or conclusive case for

one method of apportionment as fairest or most equitable. Are there other factors

that might provide additional guidance in making such an evaluation? The remaining

sections of this report examine three additional possibilities put forward by

statisticians: (1) mathematical tests different from those examined in the NAS

report; (2) standards of fairness derived from the concept of states’ representational

of Representatives assigned to the state whose average district size is the smallest (the most

over represented state). From this number subtract the number of seats assigned to the state

with the largest average district size (the most under represented state). Multiply this

remainder by the population of the most over represented state divided by the population of

the most under represented state. This number is the absolute representational surplus of

the state with the smallest average district size as compared to the state with the largest

average district size. In equation form this may be stated as follows: S=(a-b)*(A/B) where

S is the absolute representation surplus, A is the population of the over represented state, B

is the population of the under-represented state, a is the number of representatives of the

over represented state, and b is the number of representatives of the under represented state.

For further information about this test, see: Schmeckebier, Congressional Apportionment,

pp. 45-46.

number of Representatives assigned to the state whose average district size is the largest (the

most under represented state). From this number subtract the number of seats assigned to

the state with the largest average district size (the most over represented state) multiplied

by the population of the under represented state divided by the population of the over

represented state. This number is the absolute representational deficiency of the state with

the smallest average district size, as compared to the state with the largest average district

size. In equation form, this may be stated as follows: D=b-((a*B)/A) where D is the

absolute representation deficiency, A is the population of the over represented state, B is the

population of the under represented state, a is the number of representatives of the over

represented state, and b is the number of representatives of the under represented state. For

further information about this test, see Schmeckebier, Congressional Apportionment, pp. 52-

54.

“quotas”; and (3) the principles of the constitutional “great compromise” between

large and small states that resulted in the establishment of a bicameral Congress.

Alternative Kinds of Tests

As the discussion of the NAS report showed, the NAS tested each of its criteria

for evaluating apportionment methods by its effect on pairs of states. (The

descriptions of the NAS tests above stated them in terms of the highest and lowest

states for each measure, but, in fact, comparisons between all pairs of states were

used.) These pairwise tests, however, are not the only means by which different

methods of apportionment can be tested against various criteria of fairness.

For example, it is indisputable that, as the state of Montana contended in 1992,

the Dean method minimizes absolute differences in state average district populations

in the pairwise test. One of the federal government’s counter arguments, however,

was that the Dean method does not minimize such differences when all states are

considered simultaneously. The federal government proposed variance as a means

of testing apportionment formulas against various criteria of fairness.

The variance of a set of numbers is the sum of the squares of the deviations of

the individual values from the mean or average.

summarizing the degree to which individual values in a list vary from the average

(mean) of all the values in the list. High variances indicate that the values vary

greatly; low variances mean the values are similar. If all values in the lists are

identical, the variance is zero. According to this test, in other words, the smaller the

variance, the more equitable the method of apportionment.

If the variance for a Dean apportionment is compared to that of a Hill

apportionment in 1990 (using the difference between district sizes as the criterion),

the apportionment variance under Hill’s method is smaller than that under Dean’s

(see Table 4). In fact, using average district size as the criterion and variance as the

test, the variance under the Hill method is the smallest of any of the apportionment

methods discussed in this report.

the entire country and then subtract that number from each state’s average size district. This

may result in a positive or negative number. The square of this number eliminates any

negative signs. To find the total variance for a state, multiply this number by the total seats

assigned to the state. To find the variance for entire country, sum all the state variances.

Table 4. Alternate Methods for Measuring Equality

of District Sizes

Variances can be calculated, however, not only for differences in average district

size, but for each of the criteria of fairness used in pairwise tests in the 1929 NAS

report. As with those pairwise tests, different apportionment methods are evaluated

as most equitable, depending on which measure the variance is calculated for. For

example, if the criterion used for comparison is the individual share of a

Representative, the Hamilton-Vinton method proves most effective in minimizing

inequality, as measured by variance (with Webster the best of the rounding methods).

The federal government in the Massachusetts case also presented another

argument to challenge the basis for both the Montana and Massachusetts claims that

the Hill method is unconstitutional. It contended that percent difference calculations

are more fair than absolute differences, because absolute differences are not

influenced by whether they are positive or negative in direction.

Tests other than pairwise comparisons and variance can also be applied. For

example, Table 4 reports data for each method using the sum of the absolute values

(rather than the squares) of the differences between national averages and state

figures. Using this test for state differences from the national “ideal” both for

district sizes and for shares of a Representative, the Hamilton-Vinton method again

of Massachusetts, et. al. v. Mosbacher, et. al. CV NO. 91-111234 (W.D. Mass. 1991), p. 13.

calculated as follows. Each state’s average size district is subtracted from the national

“ideal size” district. (In some cases this will result in a negative number, but this calculation

uses the “absolute value” of the numbers, which always is expressed as a positive number.)

This absolute value for each state is multiplied by the number of seats the method assigns

to the state. These state totals of differences from the national ideal size are then summed

for the entire nation.

produces the smallest national totals. Of the rounding methods, again, the Webster

method minimizes both these differences.

Fairness and Quota

These examples, in which different methods best satisfy differing tests of a

variety of criteria for evaluation, serve to illustrate further the point made earlier, that

no single method of apportionment need be unambiguously the most equitable by all

measures. Each apportionment method discussed in this report has a rational basis,

and for each, there is at least one test according to which it is the most equitable. The

question of how the concept of fairness can best be defined, in the context of

evaluating an apportionment formula, remains open.

Another approach to this question begins from the observation that, if

representation were to be apportioned among the states truly according to population,

the fractional remainders would be treated as fractions rather than rounded. Each

state would be assigned its exact quota of seats, derived by dividing the national

“ideal” size district into the state’s apportionment population. There would be no

“fractional Representatives,” just fractional votes based on the states’ quotas.

Quota Representation. The Congress could weight each Representative’s

vote to account for how much his or her constituents were either over or under

represented in the House. In this way, the states’ exact quotas would be represented,

but each Representative’s vote would count differently. (This might be an easier

solution than trying to apportion seats so they crossed state lines, but it would,

however, raise other problems relating to potential inequalities of influence among

individual Representatives.

If this “quota representation” defines absolute fairness, then the concept of the

quota, rather than some statistical test, can be used as the basis of a simple concept

for judging the relative fairness of apportionment methods: a method should never

make a seat allocation that differs from a state’s exact quota by more than one seat.

Unfortunately, this concept is complicated in its application by the constitutional

requirement that each state must get one seat regardless of population size. Hence,

some modification of the quota concept is needed to account for this requirement.

One solution is the concept of fair share, which accounts for entitlements to less

than one seat by eliminating them from the calculation of quota. After all, if the

Constitution awards a seat for a fraction of less than one, then, by definition, that is

the state’s fair share of seats.

To illustrate, consider a hypothetical country with four states having populations

580, 268, 102, and 50 (thousand) and a House of 10 seats to apportion. Then the

Based on this quota, each Virginia Representative would be entitled to 1.0976 votes each

in the House. Their votes would “weigh” more than Alaska’s single Representative whose

vote would count 0.972 based on Alaska’s quota.

quotas are 5.80, 2.68, 1.02 and .50. But if each state is entitled to at least one

whole seat, then the fair share of the smallest state is 1 exactly. This leaves 9

seats to be divided among the rest. Their quotas of 9 seats are 5.49, 2.54, and

.97. Now the last of these is entitled to 1 seat, so its fair share is 1 exactly,

leaving 8 seats for the rest. Their quotas of 8 are 5.47 and 2.53. Since these are

both greater than 1, they represent the exact fractional representation that these

two states are entitled to; i.e. they are the fair shares.

Having accounted for the definitional problem of the constitutional minimum

of one seat, the revised measure is not the exact quota, but the states’ fair shares.

Which method meets the goal of not deviating by more than one seat from a state’s

fair share? No rounding method meets this test under all circumstances. Of the

methods described in this report, only the Hamilton-Vinton method always stays

within one seat of a state’s fair share. Some rounding methods are better than others

in this respect. Both the Adams and Jefferson methods nearly always produce

examples of states that get more than one seat above or below their fair shares.

Through experimentation we learn that the Dean method tends to violate this concept

approximately one percent of the time, while Webster and Hill violate it much less

than one percent of the time.

Implementing the “Great Compromise”

The framers of the Constitution (as noted earlier) created a bicameral Congress

in which representation for the states was equal in the Senate and apportioned by

population in the House. In the House, the principal means of apportionment is by

population, but each state is entitled to one Representative regardless of its

population level. Given our understanding that the “great compromise” was struck,

in part, in order to balance the interests of the smaller states with those of the larger

ones, how well do the various methods of apportionment contribute to this end?

If it is posited that the combination of factors favoring the influence of small

states encompassed in the great compromise (equal representation in the Senate, and

a one seat minimum in the House) unduly advantages the small states, then

compensatory influence could be provided to the large states in an apportionment

formula. This approach would suggest the adoption of the Jefferson method because

it significantly favors large states.

If it is posited that the influence of the small states is overshadowed by the

larger ones (perhaps because the dynamics of the electoral college focus the attention

of presidential candidates on larger states, or the increasing number of one-

Representative states — from five to seven since 1910), there are several methods

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

CRS 84-15, Sept. 30, 1984, p. 3.

55 seats to California and 33 seats to Texas when these states’ quotas (state population

divided by 1/435 of the apportionment population) are 52.45 and 32.31 respectively.

that could reduce the perceived inbalance. The Adams method favors small states

in the extreme, Dean much less so, and Hill to a small degree.

If it is posited that an apportionment method should be neutral in its application

to the states, two methods may meet this requirement. Both the Webster and

Hamilton-Vinton methods are considered to have these properties.

Conclusion

If Congress decides to revisit the matter of the apportionment formula, this

report illustrates that there could be many competing criteria from which it can

choose as a basis for decision. Among the competing mathematical tests are the pair-

wise measures proposed by the National Academy of Sciences in 1929. The federal

government proposed the statistical test of variance as an appropriate means of

computing a total for all the districts in the country in the 1992 litigation on this

matter. The plaintiffs in Massachusetts argued that variance can be computed for

different criteria than those proposed by the federal government — with different

variance measures leading to different methods.

The contention that one method or another best implements the “great

compromise” is open to much discussion. All of the competing points suggest that

Congress would be faced with difficult choices if it decided to take this issue up prior

to the 2010 Census. Which of the mathematical tests discussed in this report best

approximates the constitutional requirement that Representatives be apportioned

among the states according to their respective numbers is, arguably, a matter of

judgment — not some indisputable mathematical test.

Lawrence R. Ernst) but the evidence that the Hill method is slightly biased toward small

states is more persuasive than the criticism. See Balinski and Young, Evaluation of

Apportionment Methods, noted above.