Order Code RL31074
CRS Report for Congress
Received through the CRS Web
The House of Representatives Apportionment
Formula: An Analysis of Proposals for Change and
Their Impact on States
August 10, 2001
David C. Huckabee
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
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
The House of Representatives
Apportionment Formula:
An Analysis of Proposals for Change
and Their Impact on States
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.1
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.2
The three-judge panel’s decision came almost on the 50th anniversary of the current
formula’s enactment.3
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 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
1 See: Brookings Institution Policy Brief, Dividing the House: Why Congress Should
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.
2 Montana v. Department of Commerce, No. CV. 91-22-H-CCL.(D. Mt. Oct. 18, 1991).
U.S. District Court for the District of Montana, Helena Division.
3 55 Stat. 761, codified in 2 U.S.C. 2a, was enacted November 15, 1941.
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“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.”4
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.5
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, 2001.6
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 used in the
4 Department of Commerce v. Montana 503 U.S. 442 (1992).
5 Franklin v. Massachusetts, 505 U.S. 788 (1992). The Administrative Procedures Act
(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.
6 Utah v. Evans, No. F-2-01-CV-23: B (D. Utah, complaint filed Jan. 10, 2000).
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.
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calculations, this report describes apportionment options from which Congress could
choose and the criteria that each method satisfies.7
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.8 The Montana legal challenge added a new
perspective to the picture—determining which method comes closest to meeting the goal
of “one person, one vote.”
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
7 Representative Fithian (H.R. 1990) and Senator Lugar (S. 695) introduced bills in the 97th
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.
8 Article I, Section 3 defines both the maximum and minimum size of the House; 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 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).
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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.9
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 ....10
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.”
9 CRS Report 93-1060 GOV, Congressional Redistricting: Federal Law Controls a
State Process, by David C. Huckabee, pp. 53-54.
10 M. L. Balinski and H. P. Young, Fair Representation, 2nd ed. (Washington: Brookings
Institution Press, 2001), p. 31.
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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.11 This procedure, which might be
described as the largest fractional remainders method, was used by Congress from 1851
to 1901;12 but it was never strictly followed because changes were made in the
apportionments that were not consistent with the method.13 It has 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 population14 by the size of the House to obtain
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
11 Fair Representation, p. 21.
12 Laurence F. Schmeckebier, Congressional Apportionment (Washington: The Brookings
Institution, 1941). p. 73.
13 Fair Representation, p. 37.
14 The apportionment population is the population of the fifty states found by the Census.
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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 300.15
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.16 This 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.
15 Fair Representation, p. 38.
16 Ibid., p. 39.
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Table 1. Apportioning the House in 2001 by Simple Rounding
and Ranked Fractional Remainders (Hamilton-Vinton)
Whole
Allocation of seats
States ranked
number of
Fractional
by fractional
seats
remainders
Hamilton-
Simple
remainders
Quota
assigned
Vinton
rounding
North Dakota
0.995
1
0.99506
1
1
Virginia
10.976
10
0.97562
11
11
Maine
1.975
1
0.97500
2
2
Alaska
0.972
1
0.97215
1
1
Arizona
7.946
7
0.94600
8
8
Vermont
0.943
1
0.94271
1
1
Louisiana
6.925
6
0.92520
7
7
New Hampshire
1.914
1
0.91423
2
2
Alabama
6.896
6
0.89561
7
7
Hawaii
1.881
1
0.88058
2
2
Massachusetts
9.824
9
0.82386
10
10
New Mexico
2.819
2
0.81910
3
3
Tennessee
8.811
8
0.81060
9
9
West Virginia
2.802
2
0.80249
3
3
Florida
24.776
24
0.77601
25
25
Wyoming
0.766
1
0.76560
1
1
Georgia
12.686
12
0.68560
13
13
Missouri
8.666
8
0.66565
9
9
Colorado
6.665
6
0.66492
7
7
Nebraska
2.651
2
0.65146
3
3
Rhode Island
1.622
1
0.62247
2
2
Minnesota
7.614
7
0.61365
8
8
Ohio
17.582
17
0.58173
18
18
Iowa
4.532
4
0.53190
5
5
North Carolina
12.470
12
0.47028
13
12
Utah
3.457
3
0.45731
4
3
California
52.447
52
0.44715
52
52
Indiana
9.415
9
0.41458
9
9
Mississippi
4.410
4
0.40980
4
4
Montana
1.399
1
0.39936
1
1
Michigan
15.389
15
0.38882
15
15
New York
29.376
29
0.37617
29
29
Oklahoma
5.346
5
0.34633
5
5
Texas
32.312
32
0.31150
32
32
Wisconsin
8.302
8
0.30233
8
8
Oregon
5.300
5
0.29953
5
5
Connecticut
5.270
5
0.27015
5
5
Kentucky
6.259
6
0.25924
6
6
Illinois
19.227
19
0.22714
19
19
South Carolina
6.222
6
0.22157
6
6
Delaware
1.213
1
0.21349
1
1
Maryland
8.204
8
0.20445
8
8
South Dakota
1.170
1
0.16991
1
1
Kansas
4.164
4
0.16387
4
4
Arkansas
4.142
4
0.14209
4
4
Washington
9.133
9
0.13311
9
9
Nevada
3.095
3
0.09456
3
3
New Jersey
13.022
13
0.02160
13
13
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Whole
Allocation of seats
States ranked
number of
Fractional
by fractional
seats
remainders
Hamilton-
Simple
remainders
Quota
assigned
Vinton
rounding
Pennsylvania
19.013
19
0.01326
19
19
Idaho
2.005
2
0.00521
2
2
Total
435
413
435
433
Source: Data calculated by CRS. The “quota” is found by dividing the state population by the
national “ideal size” district (645,632 based on the 2000 Census). North Carolina and Utah receive
additional seats with the Hamilton-Vinton system even though their fractional remainders are less than
.5.
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.17
17 Balinski and Young, in Fair Representation, refer to these as divisor methods because
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 a consistent
framework to understand them all.
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Figure 1. Illustrative Rounding Points for Five Apportionment
Methods (for Two and Twenty-one Seats)
This illustration is adapted from, Balinski, M . L. and H. P. Young, Fair Representation, 2nd ed.
(Washington: Brookings Institution Press, 2001), pp. 63-65.
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.18 The key difference among these
methods is in the rule by which the 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 the intuitive basis for challenging
the Dean and Hill methods as favoring small states at the expense of the large (more
populous) states.19
18 For a detailed explanation of how apportionments are done using priority lists, see CRS
Report RL30711, The House Apportionment Formula in Theory and Practice, by David
C. Huckabee.
19 Peyton Young states that the Hill method “systematically favors the small states by 3-4
percent.” He determined this figure by first eliminating from the calculations the very small
(continued...)
CRS-10
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:
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
19 (...continued)
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.
CRS-11
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.20 The Dean 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.)
20 Expressed as a formula, the harmonic mean (H) of the numbers (A) and (B) is: H =
2*(A*B)/(A+B).
CRS-12
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.
CRS-13
Table 2. Seat Assignments in 2001 for Various House
Apportionment Formulas (Alphabetical Order)
Apportionment Method:
Current
Ranked
method:
Harm-
fractional
equal
Largest
Apportion-
Smallest
onic
remainders
pro-
Major
divisors
ment
divisors
mean
(Hamilton-
portions
fractions
(Jeffer-
ST
population
Quotaa
(Adams)
(Dean)
Vinton)
(Hill)
(Webster)
son)
AL
4,461,130
6.896
7
7
7
7
7
7
AK
628,933
0.972
1
1
1
1
1
1
AZ
5,140,683
7.946
8
8
8
8
8
8
AR
2,679,733
4.142
4
4
4
4
4
4
CA
33,930,798
52.447
50
52
52
53
53
55
CO
4,311,882
6.665
7
7
7
7
7
7
CT
3,409,535
5.270
6
5
5
5
5
5
DE
785,068
1.213
2
1
1
1
1
1
FL
16,028,890
24.776
24
25
25
25
25
26
GA
8,206,975
12.686
13
13
13
13
13
13
HI
1,216,642
1.881
2
2
2
2
2
1
ID
1,297,274
2.005
2
2
2
2
2
2
IL
12,439,042
19.227
19
19
19
19
19
20
IN
6,090,782
9.415
9
9
9
9
9
9
IA
2,931,923
4.532
5
5
5
5
5
4
KS
2,693,824
4.164
4
4
4
4
4
4
KY
4,049,431
6.259
6
6
6
6
6
6
LA
4,480,271
6.925
7
7
7
7
7
7
ME
1,277,731
1.975
2
2
2
2
2
2
MD
5,307,886
8.204
8
8
8
8
8
8
MA
6,355,568
9.824
10
10
10
10
10
10
MI
9,955,829
15.389
15
15
15
15
15
16
MN
4,925,670
7.614
8
8
8
8
8
7
MS
2,852,927
4.410
5
4
4
4
4
4
MO
5,606,260
8.666
9
9
9
9
9
9
MT
905,316
1.399
2
2
1
1
1
1
NE
1,715,369
2.651
3
3
3
3
3
2
NV
2,002,032
3.095
3
3
3
3
3
3
NH
1,238,415
1.914
2
2
2
2
2
2
NJ
8,424,354
13.022
13
13
13
13
13
13
NM
1,823,821
2.819
3
3
3
3
3
2
NY
19,004,973
29.376
28
29
29
29
29
30
NC
8,067,673
12.470
12
12
13
13
13
13
ND
643,756
0.995
1
1
1
1
1
1
OH
11,374,540
17.582
17
18
18
18
18
18
OK
3,458,819
5.346
6
5
5
5
5
5
OR
3,428,543
5.300
6
5
5
5
5
5
PA
12,300,670
19.013
19
19
19
19
19
19
RI
1,049,662
1.622
2
2
2
2
2
1
SC
4,025,061
6.222
6
6
6
6
6
6
SD
756,874
1.170
2
1
1
1
1
1
TN
5,700,037
8.811
9
9
9
9
9
9
TX
20,903,994
32.312
31
32
32
32
32
33
UT
2,236,714
3.457
4
4
4
3
3
3
VT
609,890
0.943
1
1
1
1
1
1
VA
7,100,702
10.976
11
11
11
11
11
11
CRS-14
Apportionment Method:
Current
Ranked
method:
Harm-
fractional
equal
Largest
Apportion-
Smallest
onic
remainders
pro-
Major
divisors
ment
divisors
mean
(Hamilton-
portions
fractions
(Jeffer-
ST
population
Quotaa
(Adams)
(Dean)
Vinton)
(Hill)
(Webster)
son)
WA
5,908,684
9.133
9
9
9
9
9
9
WV
1,813,077
2.802
3
3
3
3
3
2
WI
5,371,210
8.302
8
8
8
8
8
8
WY
495,304
0.766
1
1
1
1
1
1
281,424,177
a A state’s quota of Representatives is obtained by dividing the population of the fifty states by 435
to obtain a common divisor (645,632 in 2001) which is in turn divided into each state’s population.
CRS-15
Table 3. Seat Assignments in 2001 for Various House
Apportionment Formulas
(Ranked by State Population)
Apportionment Method:
Current
Ranked
method:
Harm-
fractional
equal
Largest
Apportion-
Smallest
onic
remainders
pro-
Major
divisors
ment
divisors
mean
(Hamilton-
portions
fractions
(Jeffer-
ST
population
Quotaa
(Adams)
(Dean)
Vinton)
(Hill)
(Webster)
son)
CA
33,930,798
52.450
50
52
52
53
53
55
TX
20,903,994
32.312
31
32
32
32
32
33
NY
19,004,973
29.376
28
29
29
29
29
30
FL
16,028,890
24.776
24
25
25
25
25
26
IL
12,439,042
19.227
19
19
19
19
19
20
PA
12,300,670
19.013
19
19
19
19
19
19
OH
11,374,540
17.582
17
18
18
18
18
18
MI
9,955,829
15.389
15
15
15
15
15
16
NJ
8,424,354
13.022
13
13
13
13
13
13
GA
8,206,975
12.686
13
13
13
13
13
13
NC
8,067,673
12.470
12
12
13
13
13
13
VA
7,100,702
10.976
11
11
11
11
11
11
MA
6,355,568
9.824
10
10
10
10
10
10
IN
6,090,782
9.415
9
9
9
9
9
9
WA
5,908,684
9.133
9
9
9
9
9
9
TN
5,700,037
8.811
9
9
9
9
9
9
MO
5,606,260
8.666
9
9
9
9
9
9
WI
5,371,210
8.302
8
8
8
8
8
8
MD
5,307,886
8.204
8
8
8
8
8
8
AZ
5,140,683
7.946
8
8
8
8
8
8
MN
4,925,670
7.614
8
8
8
8
8
7
LA
4,480,271
6.925
7
7
7
7
7
7
AL
4,461,130
6.896
7
7
7
7
7
7
CO
4,311,882
6.665
7
7
7
7
7
7
KY
4,049,431
6.259
6
6
6
6
6
6
SC
4,025,061
6.222
6
6
6
6
6
6
OK
3,458,819
5.346
6
5
5
5
5
5
OR
3,428,543
5.300
6
5
5
5
5
5
CT
3,409,535
5.270
6
5
5
5
5
5
IA
2,931,923
4.532
5
5
5
5
5
4
MS
2,852,927
4.410
5
4
4
4
4
4
KS
2,693,824
4.164
4
4
4
4
4
4
AR
2,679,733
4.142
4
4
4
4
4
4
UT
2,236,714
3.457
4
4
4
3
3
3
NV
2,002,032
3.095
3
3
3
3
3
3
NM
1,823,821
2.819
3
3
3
3
3
2
WV
1,813,077
2.802
3
3
3
3
3
2
NE
1,715,369
2.651
3
3
3
3
3
2
ID
1,297,274
2.005
2
2
2
2
2
2
ME
1,277,731
1.975
2
2
2
2
2
2
NH
1,238,415
1.914
2
2
2
2
2
2
HI
1,216,642
1.881
2
2
2
2
2
1
RI
1,049,662
1.622
2
2
2
2
2
1
MT
905,316
1.399
2
2
1
1
1
1
CRS-16
Apportionment Method:
Current
Ranked
method:
Harm-
fractional
equal
Largest
Apportion-
Smallest
onic
remainders
pro-
Major
divisors
ment
divisors
mean
(Hamilton-
portions
fractions
(Jeffer-
ST
population
Quotaa
(Adams)
(Dean)
Vinton)
(Hill)
(Webster)
son)
DE
785,068
1.213
2
1
1
1
1
1
SD
756,874
1.170
2
1
1
1
1
1
ND
643,756
0.995
1
1
1
1
1
1
AK
628,933
0.972
1
1
1
1
1
1
VT
609,890
0.943
1
1
1
1
1
1
WY
495,304
0.766
1
1
1
1
1
1
281,424,177
a A state’s quota of Representatives is obtained by dividing the population of the fifty states by 435
to obtain a common divisor (645,632 in 2001) which is in turn divided into each state’s population.
CRS-17
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 case21 was which formula best approximates
the “one person, one vote” principle. The apportionment concerns raised in the
Massachusetts case22 not only raised “one person, one vote” issues, but 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.23 This report predates the Supreme Court’s 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?
! The method that minimizes the difference between the largest average district size
in the country and the smallest? This criterion leads to the Dean method.
! The method that minimizes the difference in each person’s individual share of his or
her Representative by subtracting the largest such share for a state from the smallest
share? This criterion leads to the Webster method.
21 Department of Commerce v. Montana, 503 U.S. 441 (1992).
22 Franklin v. Massachusetts, 505 U.S. 788 (1992).
23 U.S. Congress, House, Committee on Post Office and Civil Service, Subcommittee on
Census and Statistics, The Decennial Population Census and Congressional
Apportionment, Appendix C: Report of National Academy of Sciences Committee on
Apportionment, 91st Cong., 1st Sess., H.Rept. 91-1314 (Washington: GPO, 1970), pp. 19-21.
CRS-18
! The method that minimizes the difference in average district sizes, or in individual
shares of a Representative, when those differences are expressed as percentages?
These criteria both lead to the Hill method.
! The method that minimizes the absolute representational surplus among states?24
This criterion leads to the Adams method.
! The method that minimizes the absolute representational deficiency among states?25
This criterion leads to the Jefferson method.
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 “quotas”; and (3) the principles of the constitutional
“great compromise” between large and small states that resulted in the establishment of a
bicameral Congress.
24 The absolute representational surplus is calculated in the following way. Take the number
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.
25 The absolute representational deficiency is calculated in the following way. Take the
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.
CRS-19
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.26 This measure is a useful way of 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.
26 In order to calculate variance for average district size, first find the ideal size district for
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.
CRS-20
Table 4. Alternate Methods for Measuring Equality
of District Sizes
Criteria for evaluation: values to be minimized
Variance
Sum of absolute values of differences
Method
Average
Individual
Average
Individual shares
district
shares
district
size
size
Adams
1,911,209,406
0.0354959
13,054,869
44.2368122
Dean
681,742,417
0.0077953
7,170,067
22.3962477
Hill (current)
661,606,402
0.0058026
7,016,021
21.3839214
Webster
665,606,402
0.0057587
6,997,789
21.2530467
Hamilton-Vinton
676,175,430
0.0057013
6,977,798
21.0633312
Jefferson
2,070,360,118
0.0112808
11,149,720
31.9326856
Bolded and Italicized numbers are the smallest for the category. The closer the values are to zero, the
closer the method comes to equalizing district sizes in the entire country. Source: CRS.
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.27
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.28
2 7 Declaration of Lawrence R. Ernst filed on behalf of the Government in Commonwealth
of Massachusetts, et. al. v. Mosbacher, et. al. CV NO. 91-111234 (W.D. Mass. 1991),
p. 13.
28 This is not a “standard” statistical test such as computing the variance. This measure is
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
(continued...)
CRS-21
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 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.29)
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.30 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.
28 (...continued)
state. These state totals of differences from the national ideal size are then summed for the
entire nation.
29 For example, Virginia’s quota of Representatives based on 2000 Census was 10.976.
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.
30 Fair Representation, p. 79.
CRS-22
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
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.31
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.32
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.33
31 Balinski, M. L. and H. P. Young, Evaluation of Apportionment Methods, Prepared
Under a Contract for the Congressional Research Service of the Library of Congress,
Contract No. CRS 84-15, Sept. 30, 1984, p. 3.
32 Ibid., p. 16.
33 Table 3 rank-orders the states by their 1990 populations. The Jefferson method awards
(continued...)
CRS-23
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 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.34
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.35
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.
33 (...continued)
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.
34 There is disagreement on this point as it pertains to the Hill method (Declaration of
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.
35 Evaluation of Apportionment Methods, p. 10-12.