The House of Representatives Apportionment Formula: An Analysis of Proposals for Change and Their Impact on States

CRS Report for Congress
The House of Representatives Apportionment
Formula: An Analysis of Proposals for Change
and Their Impact on States
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

The House of Representatives
Apportionment Formula:
An Analysis of Proposals for Change
1
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,²
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 of³
Title 2, United States Code unconstitutional and void.
The three-judge panel’s decision came almost on the 50^th anniversary of the current
formula’s enactment.⁴
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

¹ This report originally was authored by David C. Huckabee, who has retired from CRS.
² 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.
³ 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.
⁴ 55 Stat. 761, codified in 2 U.S.C. 2a, was enacted November 15, 1941.

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.”⁵
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.⁶
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, 2001.⁷
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

⁵ Department of Commerce v. Montana 503 U.S. 442 (1992).
⁶ 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.
⁷ 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.

used in the calculations, this report describes apportionment options from which
Congress could choose and the criteria that each method satisfies.⁸
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.⁹ 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.”

⁸ Representative Fithian (H.R. 1990) and Senator Lugar (S. 695) introduced bills in the 97^th
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.
⁹ 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).

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.¹⁰
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 as¹¹
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.”

¹⁰ CRS Report 93-1060 GOV, Congressional Redistricting: Federal Law Controls a State
Process, by Royce Crocker, pp. 53-54.
¹¹ M. L. Balinski and H. P. Young, Fair Representation, 2^nd ed. (Washington: Brookings
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.¹² This procedure,
which might be described as the largest fractional remainders method, was used by
Congress from 1851 to 1901;¹³ but it was never strictly followed because changes
were made in the apportionments that were not consistent with the method.¹⁴ 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 population¹⁵ 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

¹² Fair Representation, p. 21.
¹³ Laurence F. Schmeckebier, Congressional Apportionment (Washington: The Brookings
Institution, 1941). p. 73.
¹⁴ Fair Representation, p. 37.
¹⁵ The apportionment population is the population of the fifty states found by the Census.

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 but¹⁶

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.¹⁷ 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.

¹⁶ Fair Representation, p. 38.
¹⁷ Ibid., p. 39.

Table 1. Apportioning the House in 2001 by Simple Rounding
and Ranked Fractional Remainders (Hamilton-Vinton)
^Wh^ol^e^A^llo^catioⁿ^o^f^seats
^States^ran^k^e^dⁿ^u^m^{ber of}^Fr^actioⁿ^a^l
^b^y^f^r^actioⁿ^a^l^s^eats^rem^aⁱⁿ^ders^Ham^iltoⁿ^-^Sim^p^le
^rem^aⁱⁿ^ders^Q^u^ot^a^a^ssigne^d^Vin^t^oⁿ^r^o^undⁱⁿ^g
^N^o^rt^h^D^a^k^o^t^a^0.995¹⁰^.99506¹¹
^Vⁱ^rgⁱⁿⁱ^a^10.976¹⁰^0.97562¹¹¹¹
^Maiⁿ^e¹^.975¹⁰^.97500²²
^A^l^as^k^a^0.972¹⁰^.97215¹¹
^A^rⁱ^z^on^a⁷^.946⁷⁰^.94600⁸⁸
^V^e^rm^on^t⁰^.943¹⁰^.94271¹¹
^L^o^uⁱ^sⁱ^an^a⁶^.925⁶⁰^.92520⁷⁷
^N^e^w^H^a^m^p^s^hⁱ^r^e¹^.914¹⁰^.91423²²
^A^l^abam^a⁶^.896⁶⁰^.89561⁷⁷
^H^a^w^aⁱⁱ^1.881¹⁰^.88058²²
^Mas^s^ach^u^s^et^t^s^9.824⁹⁰^.82386¹⁰¹⁰
^N^e^w^Mexⁱ^co^2.819²⁰^.81910³³
^T^eⁿⁿ^e^s^s^e^e⁸^.811⁸⁰^.81060⁹⁹
^Wes^t^Vⁱ^rgⁱⁿⁱ^a^2.802²⁰^.80249³³
^F^l^ori^d^a^24.776²⁴^0.77601²⁵²⁵
^Wy^omⁱⁿ^g⁰^.766¹⁰^.76560¹¹
^G^e^orgⁱ^a^12.686¹²^0.68560¹³¹³
^Mi^s^s^o^u^rⁱ⁸^.666⁸⁰^.66565⁹⁹
^C^o^l^o^rado^6.665⁶⁰^.66492⁷⁷
^N^e^bras^k^a^2.651²⁰^.65146³³
^R^h^{ode Is}^l^aⁿ^d^1.622¹⁰^.62247²²
^Miⁿⁿ^e^s^o^t^a^7.614⁷⁰^.61365⁸⁸
^O^hⁱ^o^17.582¹⁷^0.58173¹⁸¹⁸
^Iow^a^4.532⁴⁰^.53190⁵⁵
^N^o^rth^C^a^rolⁱⁿ^a^12.470¹²^0.47028¹³¹²
^U^t^ah^3.457³⁰^.45731⁴³
^C^a^lⁱ^f^ornⁱ^a^52.447⁵²^0.44715⁵²⁵²
^In^di^an^a⁹^.415⁹⁰^.41458⁹⁹
^Mi^s^sⁱ^s^sⁱ^ppi^4.410⁴⁰^.40980⁴⁴
^Mon^t^an^a¹^.399¹⁰^.39936¹¹
^Mi^chⁱ^g^an^15.389¹⁵^0.38882¹⁵¹⁵
^N^e^w^Y^o^rk^29.376²⁹^0.37617²⁹²⁹
^O^k^l^a^h^o^m^a^5.346⁵⁰^.34633⁵⁵
^T^e^x^a^s^32.312³²^0.31150³²³²
^Wi^s^c^on^sⁱⁿ⁸^.302⁸⁰^.30233⁸⁸
^O^r^eg^on^5.300⁵⁰^.29953⁵⁵
^C^oⁿⁿ^ectⁱ^c^u^t^5.270⁵⁰^.27015⁵⁵
^K^eⁿ^t^u^c^k^y^6.259⁶⁰^.25924⁶⁶
^Illin^ois^19.227¹⁹^0.22714¹⁹¹⁹
^S^o^u^t^h^C^a^rolⁱⁿ^a⁶^.222⁶⁰^.22157⁶⁶
^D^e^l^a^w^a^re^1.213¹⁰^.21349¹¹
^Mary^l^aⁿ^d^8.204⁸⁰^.20445⁸⁸
^S^o^u^t^h^D^a^k^o^t^a^1.170¹⁰^.16991¹¹
^K^aⁿ^s^as^4.164⁴⁰^.16387⁴⁴
^A^r^k^aⁿ^s^as^4.142⁴⁰^.14209⁴⁴
^Was^hⁱⁿ^g^t^on^9.133⁹⁰^.13311⁹⁹

^Wh^ol^e^A^llo^catioⁿ^o^f^seats
^States^ran^k^e^dⁿ^u^m^{ber of}^Fr^actioⁿ^a^l
^b^y^f^r^actioⁿ^a^l^s^eats^rem^aⁱⁿ^ders^Ham^iltoⁿ^-^Sim^p^le
^rem^aⁱⁿ^ders^Q^u^ot^a^a^ssigne^d^Vin^t^oⁿ^r^o^undⁱⁿ^g
^N^e^v^a^da^3.095³⁰^.09456³³
^N^e^w^Jers^ey^13.022¹³^0.02160¹³¹³
^Penⁿ^s^y^l^v^aⁿⁱ^a^19.013¹⁹^0.01326¹⁹¹⁹
^Idah^o²^.005²⁰^.00521²²
^T^o^t^a^l⁴³⁵⁴¹³⁴³⁵⁴³³
^S^o^u^r^ce:^D^a^t^a^cal^cu^l^a^t^e^{d by}^C^R^S^.^T^h^{e “}^q^u^o^t^{a” i}^s^f^o^uⁿ^d^by^di^vⁱ^diⁿ^g^t^h^{e s}^t^at^{e popu}^l^a^tⁱ^oⁿ^by^t^h^{e n}^a^tⁱ^oⁿ^a^l
^“ⁱ^{deal s}ⁱ^{ze” dis}^t^{rict (645,632 bas}^e^{d on}^th^{e 2000 C}^eⁿ^s^u^s^).^North^C^a^rolin^a^an^{d Utah}^receiv^e^addition^a^l
^se^a^t^{s w}^{ith the}^Ha^m^ilto^n-Vintoⁿ^sy^ste^m^e^v^eⁿ^tho^{ugh the}ⁱ^r^fr^a^c^tio^na^{l r}^e^m^a^ind^e^r^s^a^r^e^le^{ss tha}ⁿ^.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.¹⁸

¹⁸ 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
(continued...)

Figure 1. Illustrative Rounding Points for Five Apportionment
Methods (for Two and Twenty-one Seats)
^T^h^is^illus^t^ra^{tion is}^a^d^a^p^te^d^f^r^om^,^Ba^lins^k^i,^{M. L}^.^a^{nd H}^.^P^.^Y^oung^,^{Fair Re}^pre^s^eⁿ^tation^{, 2}^nd^e^d^{. (}^W^a^s^hing^ton:
^Brook^ing^s^Ins^{titution P}^r^e^s^s^,^{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.¹⁹ 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

¹⁸ (...continued)
a consistent framework to understand them all.
¹⁹ 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 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.²⁰
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:

²⁰ 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
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.²¹ 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.)

²¹ Expressed as a formula, the harmonic mean (H) of the numbers (A) and (B) is: H =

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)
^A^pportⁱ^oⁿ^m^en^t^Met^h^od:
^Current
^Ra^nke^d^me^t^h^o^d^:
^Harm^-^f^r^actioⁿ^a^l^equa^l^La^r^g^e^s^t
^A^pportⁱ^oⁿ^-^Sm^allest^onⁱ^c^rem^aⁱⁿ^ders^pro^-^Maj^o^r^di^vⁱ^s^o^rs
^meⁿ^t^a^di^vⁱ^s^o^rs^m^ean⁽^H^am^iltoⁿ^-^po^rt^io^ns^f^r^actioⁿ^s^(Jef^f^e^r-
^ST^popu^l^a^tⁱ^oⁿ^Q^u^o^t^a^(A^dam^s⁾^(Dean⁾^Vin^t^oⁿ⁾⁽^H^ill)^(W^ebs^t^er)^soⁿ⁾
^A^L^4,461,130^6.896⁷⁷⁷⁷⁷⁷
^A^K^628,933^0.972¹¹¹¹¹¹
^A^Z^5,140,683^7.946⁸⁸⁸⁸⁸⁸
^A^R^2,679,733^4.142⁴⁴⁴⁴⁴⁴
^C^A^33,930,798^52.447⁵⁰⁵²⁵²⁵³⁵³⁵⁵
^C^O^4,311,882^6.665⁷⁷⁷⁷⁷⁷
^C^T^3,409,535^5.270⁶⁵⁵⁵⁵⁵
^D^E^785,068^1.213²¹¹¹¹¹
^F^L^16,028,890^24.776²⁴²⁵²⁵²⁵²⁵²⁶
^G^A^8,206,975^12.686¹³¹³¹³¹³¹³¹³
^H^I^1,216,642^1.881²²²²²¹
^ID^1,297,274^2.005²²²²²²
^IL^12,439,042^19.227¹⁹¹⁹¹⁹¹⁹¹⁹²⁰
^IN^6,090,782^9.415⁹⁹⁹⁹⁹⁹
^IA^2,931,923^4.532⁵⁵⁵⁵⁵⁴
^K^S^2,693,824^4.164⁴⁴⁴⁴⁴⁴
^K^Y^4,049,431^6.259⁶⁶⁶⁶⁶⁶
^L^A^4,480,271^6.925⁷⁷⁷⁷⁷⁷
^ME^1,277,731^1.975²²²²²²
^MD^5,307,886^8.204⁸⁸⁸⁸⁸⁸
^MA^6,355,568^9.824¹⁰¹⁰¹⁰¹⁰¹⁰¹⁰
^MI^9,955,829^15.389¹⁵¹⁵¹⁵¹⁵¹⁵¹⁶
^MN^4,925,670^7.614⁸⁸⁸⁸⁸⁷
^MS^2,852,927^4.410⁵⁴⁴⁴⁴⁴
^MO^5,606,260^8.666⁹⁹⁹⁹⁹⁹
^MT^905,316^1.399²²¹¹¹¹
^N^E^1,715,369^2.651³³³³³²
^N^V^2,002,032^3.095³³³³³³
^N^H^1,238,415^1.914²²²²²²
^N^J^8,424,354^13.022¹³¹³¹³¹³¹³¹³
^N^M^1,823,821^2.819³³³³³²
^N^Y^19,004,973^29.376²⁸²⁹²⁹²⁹²⁹³⁰
^N^C^8,067,673^12.470¹²¹²¹³¹³¹³¹³
^N^D^643,756^0.995¹¹¹¹¹¹
^O^H^11,374,540^17.582¹⁷¹⁸¹⁸¹⁸¹⁸¹⁸
^O^K^3,458,819^5.346⁶⁵⁵⁵⁵⁵
^O^R^3,428,543^5.300⁶
^PA^12,300,670^19.013¹⁹¹⁹¹⁹¹⁹¹⁹¹⁹
^R^I^1,049,662^1.622²²²²²¹
^S^C^4,025,061^6.222⁶⁶⁶⁶⁶⁶
^S^D^756,874^1.170²¹¹¹¹¹
^T^N^5,700,037^8.811⁹⁹⁹⁹⁹⁹
^T^X^20,903,994^32.312³¹³²³²³²³²³³
^U^T^2,236,714^3.457⁴⁴⁴³³³

^A^pportⁱ^oⁿ^m^en^t^Met^h^od:
^Current
^Ra^nke^d^me^t^h^o^d^:
^Harm^-^f^r^actioⁿ^a^l^equa^l^La^r^g^e^s^t
^A^pportⁱ^oⁿ^-^Sm^allest^onⁱ^c^rem^aⁱⁿ^ders^pro^-^Maj^o^r^di^vⁱ^s^o^rs
^meⁿ^t^a^di^vⁱ^s^o^rs^m^ean⁽^H^am^iltoⁿ^-^po^rt^io^ns^f^r^actioⁿ^s^(Jef^f^e^r-
^ST^popu^l^a^tⁱ^oⁿ^Q^u^o^t^a^(A^dam^s⁾^(Dean⁾^Vin^t^oⁿ⁾⁽^H^ill)^(W^ebs^t^er)^soⁿ⁾
^V^T^609,890^0.943¹¹¹¹¹¹
^V^A^7,100,702^10.976¹¹¹¹¹¹¹¹¹¹¹¹
^WA^5,908,684^9.133⁹⁹⁹⁹⁹⁹
^WV^1,813,077^2.802³³³³³²
^WI^5,371,210^8.302⁸⁸⁸⁸⁸⁸
^WY^495,304^0.766¹¹¹¹¹¹
^281,424,177
^a^A^s^t^at^e’^s^qu^ot^{a of}^R^e^pres^en^t^a^tⁱ^v^esⁱ^s^obt^aiⁿ^e^d^by^di^vⁱ^diⁿ^g^t^h^{e popu}^l^a^tⁱ^oⁿ^of^t^h^{e f}ⁱ^f^t^y^s^t^at^es^by⁴³⁵
^{to obtain}^{a com}^m^oⁿ^divⁱ^s^o^{r (645,632}ⁱⁿ²⁰⁰¹⁾^w^h^ich^isⁱⁿ^tu^rn^divⁱ^{ded in}^{to each}^s^t^ate’^s^popu^lation^.

Table 3. Seat Assignments in 2001 for Various House
Apportionment Formulas
(Ranked by State Population)
^A^pportⁱ^oⁿ^m^en^t^Met^h^od:
^Current
^Ra^nke^d^me^t^h^o^d^:
^Harm^-^f^r^actioⁿ^a^l^equa^l^La^r^g^e^s^t
^A^pportⁱ^oⁿ^-^Sm^allest^onⁱ^c^rem^aⁱⁿ^ders^pro^-^Maj^o^r^di^vⁱ^s^o^rs
^meⁿ^t^a^di^vⁱ^s^o^rs^m^ean⁽^H^am^iltoⁿ^-^po^rt^io^ns^f^r^actioⁿ^s^(Jef^f^e^r-
^ST^popu^l^a^tⁱ^oⁿ^Q^u^o^t^a^(A^dam^s⁾^(Dean⁾^Vin^t^oⁿ⁾⁽^H^ill)^(W^ebs^t^er)^soⁿ⁾
^C^A^33,930,798^52.450⁵⁰⁵²⁵²⁵³⁵³⁵⁵
^T^X^20,903,994^32.312³¹³²³²³²³²³³
^N^Y^19,004,973^29.376²⁸²⁹²⁹²⁹²⁹³⁰
^F^L^16,028,890^24.776²⁴²⁵²⁵²⁵²⁵²⁶
^IL^12,439,042^19.227¹⁹¹⁹¹⁹¹⁹¹⁹²⁰
^PA^12,300,670^19.013¹⁹¹⁹¹⁹¹⁹¹⁹¹⁹
^O^H^11,374,540^17.582¹⁷¹⁸¹⁸¹⁸¹⁸¹⁸
^MI^9,955,829^15.389¹⁵¹⁵¹⁵¹⁵¹⁵¹⁶
^N^J^8,424,354^13.022¹³¹³¹³¹³¹³¹³
^G^A^8,206,975^12.686¹³¹³¹³¹³¹³¹³
^N^C^8,067,673^12.470¹²¹²¹³¹³¹³¹³
^V^A^7,100,702^10.976¹¹¹¹¹¹¹¹¹¹¹¹
^MA^6,355,568^9.824¹⁰¹⁰¹⁰¹⁰¹⁰¹⁰
^IN^6,090,782^9.415⁹⁹⁹⁹⁹⁹
^WA^5,908,684^9.133⁹⁹⁹⁹⁹⁹
^T^N^5,700,037^8.811⁹⁹⁹⁹⁹⁹
^MO^5,606,260^8.666⁹⁹⁹⁹⁹⁹
^WI^5,371,210^8.302⁸⁸⁸⁸⁸⁸
^MD^5,307,886^8.204⁸⁸⁸⁸⁸⁸
^A^Z^5,140,683^7.946⁸⁸⁸⁸⁸⁸
^MN^4,925,670^7.614⁸⁸⁸⁸⁸⁷
^L^A^4,480,271^6.925⁷⁷⁷⁷⁷⁷
^A^L^4,461,130^6.896⁷⁷⁷⁷⁷⁷
^C^O^4,311,882^6.665⁷⁷⁷⁷⁷⁷
^K^Y^4,049,431^6.259⁶⁶⁶⁶⁶⁶
^S^C^4,025,061^6.222⁶⁶⁶⁶⁶⁶
^O^K^3,458,819^5.346⁶⁵⁵⁵⁵⁵
^O^R^3,428,543^5.300⁶
^C^T^3,409,535^5.270⁶⁵⁵⁵⁵⁵
^IA^2,931,923^4.532⁵⁵⁵⁵⁵⁴
^MS^2,852,927^4.410⁵⁴⁴⁴⁴
^K^S^2,693,824^4.164⁴⁴⁴⁴⁴⁴
^A^R^2,679,733^4.142⁴⁴⁴⁴⁴⁴
^U^T^2,236,714^3.457⁴⁴⁴³³³
^N^V^2,002,032^3.095³³³³³³
^N^M^1,823,821^2.819³³³³³²
^WV^1,813,077^2.802³³³³³²
^N^E^1,715,369^2.651³³³³³²
^ID^1,297,274^2.005²²²²²²
^ME^1,277,731^1.975²²²²²²
^N^H^1,238,415^1.914²²²²²²
^H^I^1,216,642^1.881²²²²²¹

^A^pportⁱ^oⁿ^m^en^t^Met^h^od:
^Current
^Ra^nke^d^me^t^h^o^d^:
^Harm^-^f^r^actioⁿ^a^l^equa^l^La^r^g^e^s^t
^A^pportⁱ^oⁿ^-^Sm^allest^onⁱ^c^rem^aⁱⁿ^ders^pro^-^Maj^o^r^di^vⁱ^s^o^rs
^meⁿ^t^a^di^vⁱ^s^o^rs^m^ean⁽^H^am^iltoⁿ^-^po^rt^io^ns^f^r^actioⁿ^s^(Jef^f^e^r-
^ST^popu^l^a^tⁱ^oⁿ^Q^u^o^t^a^(A^dam^s⁾^(Dean⁾^Vin^t^oⁿ⁾⁽^H^ill)^(W^ebs^t^er)^soⁿ⁾
^R^I^1,049,662^1.622²²²²²¹
^MT^905,316^1.399²²¹¹¹¹
^D^E^785,068^1.213²¹¹¹¹¹
^S^D^756,874^1.170²
^N^D^643,756^0.995¹¹¹¹¹¹
^A^K^628,933^0.972¹¹¹¹¹¹
^V^T^609,890^0.943¹¹¹¹¹¹
^WY^495,304^0.766¹¹¹¹¹¹
^281,424,177
^a^A^s^t^at^e’^s^qu^ot^{a of}^R^e^pres^en^t^a^tⁱ^v^esⁱ^s^obt^aiⁿ^e^{d by}^di^vⁱ^diⁿ^g^t^h^{e popu}^l^a^tⁱ^oⁿ^of^t^h^e^fⁱ^f^t^y^s^t^at^es^by⁴³⁵
^{to obtain}^{a com}^m^oⁿ^divⁱ^s^o^{r (645,632 in}^{2001) w}^h^ich^isⁱⁿ^tu^rn^divⁱ^{ded in}^{to each}^s^t^ate’^s^popu^lation^.

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 case²² was which formula best
approximates the “one person, one vote” principle. The apportionment concerns
raised in the Massachusetts case²³ 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.²⁴ 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

²² Department of Commerce v. Montana, 503 U.S. 441 (1992).
²³ Franklin v. Massachusetts, 505 U.S. 788 (1992).
²⁴ 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^st^st
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.
^!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?²⁵ This criterion leads to the Adams method.
^!The method that minimizes the absolute representational deficiency
among states?²⁶ 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

²⁵ 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.
²⁶ 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.

“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.²⁷ 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.

²⁷ 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.

Table 4. Alternate Methods for Measuring Equality
of District Sizes
^Cri^t^e^ri^{a f}^o^{r eval}^u^a^tⁱ^oⁿ^:^val^u^{es t}^o^b^e^mⁱⁿⁱ^mⁱ^zed
^V^a^ri^an^ce^S^u^m^o^f^ab^so^l^u^t^e^val^u^{es o}^f^dⁱ^f^f^e^ren^ces
^Me^thod^A^v^{erage d}ⁱ^st^ri^ct^I^ndivⁱ^dua^l^A^v^{erage d}ⁱ^st^ri^ct^I^ndivⁱ^dua^{l s}^h^a^r^e^s
^size^sh^ares^size
^A^d^a^m^s¹^,911,209,406^0.0354959^13,054,869^44.2368122
^D^e^aⁿ^681,742,417^0.0077953^7,170,067^22.3962477
^{Hill (c}^urre^nt)^661,606,402^0.0058026^7,016,021^21.3839214
^W^e^bs^te^r^665,606,402^0.0057587^6,997,789^21.2530467
^H^a^m^ilton-^Vinton^676,175,430^0.0057013^6,977,798^21.0633312
^J^e^f^f^e^r^s^on^{2,070,360,118}^0.0112808^11,149,720^31.9326856
^Bolde^d^{and Italic}^ize^dⁿ^u^m^b^e^{rs are t}^h^{e sm}^al^l^e^st^f^o^{r t}^h^{e cat}^ego^r^y^.^T^h^{e cl}^o^s^er^t^h^e^val^u^{es are t}^o^zero^,^t^h^{e cl}^o^s^{er t}^h^e
^m^e^{thod c}^o^m^e^s^{to e}^qua^liz^ing^dis^t^rⁱ^c^t^sⁱ^z^e^s^{in the}^eⁿ^tir^e^c^ountr^y^{. Sour}^c^e^:^C^R^S^.
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.²⁸
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 state²⁹
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

²⁸ 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.
²⁹ 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 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.³⁰)
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 never³¹
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

³⁰ 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.
³¹ Fair Representation, p. 79.

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 these³²
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.³³
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.³⁴
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

³² 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.
³³ Ibid., p. 16.
³⁴ Table 3 rank-orders the states by their 1990 populations. The Jefferson method awards
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.³⁵
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.³⁶
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.

³⁵ 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.
³⁶ Evaluation of Apportionment Methods, p. 10-12.