The U.S. House of Representatives
Apportionment Formula in Theory and
Practice
Royce Crocker
Specialist in American National Government
August 4, 2010
Congressional Research Service
7-5700
www.crs.gov
R41357
CRS Report for Congress
P
repared for Members and Committees of Congress

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The U.S. House of Representatives Apportionment Formula in Theory and Practice

Summary
At the end of 2010, based on the results of the 2010 Census, the number of seats allocated to each
state for the House of Representatives will be determined. This allocation likely will determine
representation to the House for the next five Congresses.
The Constitution requires that states be represented in the House of Representatives in accord
with their population. It also requires that each state have at least one Representative, and that
there be no more than one Representative for every 30,000 persons. For the 2000 apportionment,
this could have meant a House of Representatives as small as 50 or as large as 9,380
Representatives.
Apportioning seats in the House of Representatives among the states in proportion to state
population as required by the Constitution appears on the surface to be a simple task. In fact,
however, the Constitution presented Congress with issues that provoked extended and recurring
debate. How many Representatives should the House comprise? How populous should
congressional districts be? What is to be done with the practically inevitable fractional
entitlement to a House seat that results when the calculations of proportionality are made? How is
fairness of apportionment to be best preserved? Apportioning the House can be viewed as a
system with four main variables: (1) the size of the House, (2) the population of the states, (3) the
number of states, and (4) the method of apportionment.
Over the years since the ratification of the Constitution, the number of Representatives has varied,
but in 1941 Congress resolved the issue by fixing the size of the House at 435 Members. How to
apportion those 435 seats, however, continued to be an issue because of disagreement over how
to handle fractional entitlements to a House seat in a way that both met constitutional and
statutory requirements and minimized inequity.
The intuitive method of apportionment is to divide the United States population by 435 to obtain
an average number of persons represented by a Member of the House. This is sometimes called
the ideal size congressional district. Then a state’s population is divided by the ideal size to
determine the number of Representatives to be allocated to that state. The quotient will be a
whole number plus a remainder—say 14.489326. What is Congress to do with the 0.489326
fractional entitlement? Does the state get 14 or 15 seats in the House? Does one discard the
fractional entitlement? Does one round up at the arithmetic mean of the two whole numbers? At
the geometric mean? At the harmonic mean? Congress has used, or at least considered, several
methods over the years.
Every method Congress has used or considered has its advantages and disadvantages, and none
has been exempt from criticism. Under current law, however, seats are apportioned using the
equal proportions method, which is not without its critics. Some charge that the equal proportions
method is biased toward small states. They urge Congress to adopt either the major fractions or
the Hamilton-Vinton method as more equitable alternatives. A strong mathematical case can be
made for either equal proportions or major fractions. Deciding between them is a policy matter
based on whether minimizing the differences in district sizes in absolute terms (through major
fractions) or proportional terms (through equal proportions) is most preferred by Congress.

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The U.S. House of Representatives Apportionment Formula in Theory and Practice

Contents
The U.S. House of Representatives Apportionment Formula in Theory and Practice .................... 1
Introduction .......................................................................................................................... 1
Constitutional and Statutory Requirements ............................................................................ 2
The Apportionment Formula ................................................................................................. 3
The Formula in Theory ................................................................................................... 3
The Formula in Practice: Deriving the Apportionment from a Table of “Priority
Values” ........................................................................................................................ 4
Challenges to the Current Formula ........................................................................................ 8
Equal Proportions or Major Fractions: An Analysis ............................................................. 10
The Case for Major Fractions ........................................................................................ 10
The Case for Equal Proportions..................................................................................... 12

Tables
Table 1. Multipliers for Determining Priority Values for Apportioning the House by the
Equal Proportions Method........................................................................................................ 5
Table 2. Priority Rankings for Assigning Thirty Seats in a Hypothetical Three-State
House Delegation..................................................................................................................... 6
Table 3. Rounding Points for Assigning Seats Using the Equal Proportions Method of
Apportionment ......................................................................................................................... 9
Table A-1. 2000 Priority List for Apportioning Seats to the House of Representatives................ 15

Appendixes
Appendix. 2000 Priority List for Apportioning Seats to the House of Representatives................ 15

Contacts
Author Contact Information ...................................................................................................... 27

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The U.S. House of Representatives Apportionment Formula in Theory and Practice

The U.S. House of Representatives Apportionment
Formula in Theory and Practice1
Introduction
One of the fundamental issues before the framers at the Constitutional Convention in 1787 was
the allocation of representation in Congress between the smaller and larger states.2 The solution
ultimately adopted, known as the Great (or Connecticut) Compromise, resolved the controversy
by creating a bicameral Congress with states represented equally in the Senate, but in proportion
to population in the House.
The Constitution provided the first apportionment of House seats: 65 Representatives were
allocated among the states based on the framers’ estimates of how seats might be apportioned
following a census.3 House apportionments thereafter were to be based on Article 1, section 2, as
modified by the Fourteenth Amendment:
Amendment XIV, section 2. Representatives shall be apportioned among the several States
according to their respective numbers....
Article 1, section 2. The number of Representatives shall not exceed one for every thirty
Thousand, but each State shall have at least one Representative....
From its beginning in 1789, Congress was faced with questions about how to apportion the House
of Representatives—questions that the Constitution did not answer. How populous should a
congressional district be on average? How many Representatives should the House comprise?
Moreover, no matter how one specified the ideal population of a congressional district or the
number of Representatives in the House, a state’s ideal apportionment would, as a practical
matter, always be either a fraction, or a whole number and a fraction—say, 14.489326. Thus,
another question was whether that state would be apportioned 14 or 15 representatives?
Consequently, these two major issues dominated the apportionment debate: how populous a
congressional district ought to be (later re-cast as how large the House ought to be), and how to
treat fractional entitlements to Representatives.4

1 A similar, previous CRS report was authored by David C. Huckabee, who retired in 2005. While the current report is
modified by the current author, Mr. Huckabee’s contribution, in a large part, remains. Of course, any errors that may
appear are due solely to the current author.
2 In part, this debate over the apportionment of power in the early years of this country came from the 10-year
experience with the unicameral congress provided for under the Articles of Confederation, which assigned one vote to
each state delegation in Congress. For a thorough discussion, see Charles A. Kromkowski, Recreating the American
Republic, (Cambridge University Press, Cambridge, U.K., 2002), esp., pp. 261-307.
3 A major controversy occurred even over the fixed, short-term apportionment of seats among the delegates at the
Constitutional Convention. See Kromkowski, pp. 287-294.
4 Thomas Jefferson recommended discarding the fractions. Daniel Webster and others argued that Jefferson’s method
was unconstitutional because it discriminated against small states. Webster argued that an additional Representative
should be awarded to a state if the fractional entitlement was 0.5 or greater—a method that decreased the size of the
house by 17 Members in 1832. Congress subsequently used a “fixed ratio” method proposed by Rep. Samuel Vinton
following the census of 1850 through 1900, but this method led to the paradox that Alabama lost a seat even though the
size of the House was increased in 1880. Subsequently, mathematician W.F. Willcox proposed the “major fractions”
(continued...)
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The U.S. House of Representatives Apportionment Formula in Theory and Practice

The questions of how populous a congressional district should be and how many Representatives
should constitute the House have received little attention since the number of Representatives was
last increased from 386 to 435 after the 1910 Census.5 The problem of fractional entitlement to
Representatives, however, continued to be troublesome. Various methods were considered and
some were tried, each raising questions of fundamental fairness. The issue of fairness could not
be perfectly resolved: inevitable fractional entitlements and the requirement that each state have
at least one representative lead to inevitable disparities among the states’ average congressional
district populations. Congress, which sought an apportionment method that would minimize those
disparities, continued this debate until 1941, when it enacted the “equal proportions” method—
the apportionment method still in use today (for a full explanation of this method, see below).
In light of the lengthy debate on apportionment, this report has four major purposes:
1. summarize the constitutional and statutory requirements governing
apportionment;
2. explain how the current apportionment formula works in theory and in practice;
3. summarize challenges to it on grounds of inequity; and
4. explain the reasoning underlying the choice of the equal proportions method over
its chief alternative, the method of major fractions.
Constitutional and Statutory Requirements
The process of apportioning seats in the House is constrained both constitutionally and statutorily.
As noted previously, the Constitution defines both the maximum and minimum size of the House.
There can be no fewer than one Representative per state, and no more than one for every 30,000
persons.6
The Apportionment Act of 1941, in addition to specifying the apportionment method, sets the
House size at 435, requires an apportionment every 10 years, and mandates administrative
procedures for apportionment. The President is required to transmit to Congress “a statement

(...continued)
method, which was used following the census of 1910. This method, too, had its critics; and in 1921 Harvard
mathematician E.V. Huntington proposed the “equal proportions” method and developed formulas and computational
tables for all of the other known, mathematically valid apportionment methods. A committee of the National Academy
of Sciences conducted an analysis of each of those methods—smallest divisors, harmonic mean, equal proportions,
major fractions, and greatest divisors—and recommended that Congress adopt Huntington’s equal proportions method.
For a review of this history, see U.S. Congress, House, Committee on Post Office and Civil Service, Subcommittee on
Census and Statistics, The Decennial Population Census and Congressional Apportionment, 91st Congress, 2nd session.
H. Report 91-1314 (Washington: GPO, 1970), Appendix B, pp. 15-18. Also, see Michel L. Balinski and H. Peyton
Young, Fair Representation, 2nd edition, (Brookings Institution Press, Washington, 2001).
5 Article I, Section 2 defines both the maximum and minimum size of the House, but the actual House size is set by
law. There can be no fewer than one Representative per state, and no more than one for every 30,000 persons. Thus, the
House after 2000 could have been as small as 50 and as large as 9,380 Representatives.
6The actual language in of Article 1, section 2 pertaining to this minimum size reads as follows: “The number of
Representatives shall not exceed one for every thirty Thousand, but each State shall have at least one Representative.”
This clause is sometime misread to be a requirement that districts can be no larger than 30,000 persons, rather than as it
should be read, as a minimum-size population requirement.
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showing the whole number of persons in each state” and the resulting seat allocation within one
week after the opening of the first regular session of Congress following the census.7
The Census Bureau has been assigned the responsibility of computing the apportionment. As a
matter of practice, the Director of the Bureau reports the results of the apportionment on
December 31 of the census year. Once received by Congress, the Clerk of the House of
Representatives is charged with the duty of sending to the governor of each state a “certificate of
the number of Representatives to which such state is entitled” within 15 days of receiving notice
from the President.8
The Apportionment Formula
The Formula in Theory
An intuitive way to apportion the House is through simple rounding (a method never adopted by
Congress). First, the U.S. apportionment population9 is divided by the total number of seats in the
House (e.g., in 2000, 281,424,177 divided by 435) to identify the “ideal” sized congressional
district (646,952 in 2000). Then, each state’s population is divided by the “ideal” district
population. In most cases this will result in a whole number and a fractional remainder, as noted
earlier. Each state will definitely receive seats equal to the whole number, and the fractional
remainders will either be rounded up or down (at the .5 “rounding point”).
There are two fundamental problems with using simple rounding for apportionment, given a
House of fixed size. First, it is possible that some state populations might be so small that they
would be “entitled” to less than half a seat. Yet, the Constitution requires that every state must
have at least one seat in the House. Thus, a method that relies entirely on rounding will not
comply with the Constitution if there are states with very small populations. Second, even a
method that assigns each state its constitutional minimum of one seat, and otherwise relies on
rounding at the .5 rounding point, might require a “floating” House size because rounding at .5
could result in either fewer or more than 435 seats. Thus, this intuitive way to apportion fails
because, by definition, it does not take into account the constitutional requirement that every state
have at least one seat in the House and the statutory requirement that the House size be fixed at
435.
The current apportionment method (the method of equal proportions established by the 1941 act)
satisfies the constitutional and statutory requirements. Although an equal proportions
apportionment is not normally computed in the theoretical way described below, the method can
be understood as a modification of the rounding scheme described above.
First, the “ideal” sized district is found (by dividing the apportionment population by 435) to
serve as a “trial” divisor.

7 55 Stat. 761. (1941) Sec. 22 (a). [Codified in 2 U.S.C. 2(a).] In other words, after the 2010 Census, this report is due
in January 2010. Interestingly, while the Constitution requires a census every ten years, it does not require that an
apportionment of seats to the House of Representatives must occur. This became a statutory requirement with the
passage of the Apportionment Act of 1941.
8 Ibid., Sec. 22 (b).
9 The apportionment population is the population of the 50 states. It excludes the population of the District of Columbia
and U.S. territories and possessions.
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Then each state’s apportionment population is divided by the “ideal” district size to determine its
number of seats. Rather than rounding up any remainder of .5 or more, and down for less than .5,
however, equal proportions rounds at the geometric mean of any two successive numbers. A
geometric mean of two numbers is the square root of the product of the two numbers.10 If using
the “ideal” sized district population as a divisor does not yield 435 seats, the divisor is adjusted
upward or downward until rounding at the geometric mean will result in 435 seats.
For example, for the 2000 apportionment, the “ideal” size district of 646,952 had to be adjusted
downward to between 645,684 and 645,93011 to produce a 435-Member House. Because the
divisor is adjusted so that the total number of seats will equal 435, the problem of the “floating”
House size is solved. The constitutional requirement of at least one seat for each state is met by
assigning each state one seat automatically regardless of its population size.
The Formula in Practice: Deriving the Apportionment from a Table of “Priority
Values”
Although the process of determining an apportionment through a series of trials using divisions
near the “ideal” sized district as described above works, it is inefficient because it requires a
series of calculations using different divisors until the 435 total is reached. Accordingly, the
Census Bureau determines apportionment by computing a “priority” list of state claims to each
seat in the House.
During the early twentieth century, Walter F. Willcox, a Cornell University mathematician,
determined that if the rounding points used in an apportionment method are divided into each
state’s population (the mathematical equivalent of multiplying the population by the reciprocal of
the rounding point), the resulting numbers can be ranked in a priority list for assigning seats in the
House.12
Such a priority list does not assume a fixed House size because it ranks each of the states’ claims
to seats in the House so that any size House can be chosen easily without the necessity of
extensive recomputations.13

10 The geometric mean of 1 and 2 is the square root of 2, which is 1.4142. The geometric mean of 2 and 3 is the square
root of 6, which is 2.4495. Geometric means are computed for determining the rounding points for the size of any
state’s delegation size. Equal proportions rounds at the geometric mean (which varies) rather than the arithmetic mean
(which is always halfway between any pair of numbers). Thus, a state which would be entitled to 10.4871 seats before
rounding will be rounded down to 10 because the geometric mean of 10 and 11 is 10.4881. The rationale for choosing
the geometric mean rather than the arithmetic mean as the rounding point is discussed below in the section analyzing
the equal proportions and major fractions formulas.
11 Any number in this range divided into each state’s population and rounded at the geometric mean will produce a 435-
seat House.
12 U.S. Congress, House Committee on Post Office and Civil Service, Subcommittee on the Census and Statistics, The
Decennial Population Census and Congressional Apportionment, 91st Congress, 2nd session, H. Report 91-1814,
(Washington: GPO, 1970), p. 16.
13 The 435 limit on the size of the House is a statutory requirement. The House size was first fixed at 435 by the
Apportionment Act of 1911 (37 Stat. 13). The Apportionment Act of 1929 (46 Stat. 26), as amended by the
Apportionment Act of 1941 (54 Stat. 162), provided for “automatic reapportionment” rather than requiring the
Congress to pass a new apportionment law each decade. This requirement to “automatically reapportion” every 10
years was needed because the Constitution, ironically, while requiring a census every 10 years makes no such
requirement for apportionments. Thus, the fact that no apportionment was carried out after the 1920 census in no way
violated the Constitution or any statutory requirement at the time. By authority of section 9 of PL 85-508 (72 Stat. 345)
(continued...)
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The traditional method of constructing a priority list to apportion seats by the equal proportions
method involves first computing the reciprocals14 of the geometric means (the “rounding points”)
between every pair of consecutive whole numbers (representing the seats to be apportioned). It is
then possible to multiply by decimals rather than divide by fractions (the former being a
considerably easier task). For example, the reciprocal of the geometric mean between 1 and 2
(1.41452) is 1/1.414452 or .70710678, which becomes the “multiplier” for the priorities for
rounding to the second seat for each state. These reciprocals for all pairs (1 to 2, 2 to 3, 3 to 4,
etc.) are computed for each “rounding point.” They are then used as multipliers to construct the
“priority list.” Table 1, below, provides a list of multipliers used to calculate the “priority values”
for each state in an equal proportions apportionment, allowing for the allocation of up to 60 seats
to each state.
In order to construct the “priority list,” each state’s apportionment population is multiplied by
each of the multipliers. The resulting products are ranked in order to show each state’s claim to
seats in the House. For example, (see Table 2, below) assume that there are three states in the
Union (California, New York, and Florida) and that the House size is set at 30 Representatives.
The first seat for each state is assigned by the Constitution; so the remaining twenty-seven seats
must be apportioned using the equal proportions formula. The 2000 apportionment populations
for these states were 33,930,798 for California, 19,004,973 for New York, and 16,028,890 for
Florida.
Once the priority values are computed, they are ranked with the highest value first. The resulting
ranking is numbered and seats are assigned until the total is reached. By using the priority
rankings instead of the rounding procedures described earlier in this paper under “The Formula in
Theory,” it is possible to see how an increase or decrease in the House size will affect the
allocation of seats without the necessity of additional calculations.
Table 1. Multipliers for Determining Priority Values for Apportioning the House by
the Equal Proportions Method
Seat
Seat
Seat
Assignment Multipliera
Assignment Multipliera
Assignment Multipliera
1 Constitution 21 0.04879500 41 0.02469324
2 0.70710678 22 0.04652421 42 0.02409813
3 0.40824829 23 0.04445542 43 0.02353104
4 0.28867513 24 0.04256283 44 0.02299002
5 0.22360680 25 0.04082483 45 0.02247333
6 0.18257419 26 0.03922323 46 0.02197935
7 0.15430335 27 0.03774257 47 0.02150662
8 0.13363062 28 0.03636965 48 0.02105380
9 0.11785113 29 0.03509312 49 0.02061965

(...continued)
and section 8 of PL 86-3 (73 Stat. 8), which admitted Alaska and Hawaii to statehood, the House size was temporarily
increased to 437 until the reapportionment resulting from the 1960 Census when it returned to 435.
14 A reciprocal of a number is that number divided into one.
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Seat
Seat
Seat
Assignment
Multipliera
Assignment
Multipliera
Assignment
Multipliera
10 0.10540926 30 0.03390318 50 0.02020305
11 0.09534626 31 0.03279129 51 0.01980295
12 0.08703883 32 0.03175003 52 0.01941839
13 0.08006408 33 0.03077287 53 0.01904848
14 0.07412493 34 0.02985407 54 0.01869241
15 0.06900656 35 0.02898855 55 0.01834940
16 0.06454972 36 0.02817181 56 0.01801875
17 0.06063391 37 0.02739983 57 0.01769981
18 0.05716620 38 0.02666904 58 0.01739196
19 0.05407381 39 0.02597622 59 0.01709464
20 0.05129892 40 0.02531848 60 0.01680732
a. Table by CRS, calculated by determining the reciprocal of the geometric mean of successive numbers,
1/√n(n-1), where “n” is the number of seats to be allocated to the state.
More specifically, for this example in Table 2, the computed priority values (column six) for each
of the three states are ordered from largest to smallest. By constitutional provision, seats one to
three are given to each state. The next determination is the fourth seat in the hypothesized
chamber. California’s claim to a second seat, based on its priority value, is 23,992,697.36
(0.70710681 x 33,930,798), while New York’s claim to a second seat is 13,438,545.28
(0.70710681 x 19,004,973), and Florida’s claim to a second seat is 11,334,136.81 (0.70710681 x
16,028890). Based on the priority values, California has the highest claim for its second seat and
is allocated the fourth seat in the hypothesized chamber.
Table 2. Priority Rankings for Assigning Thirty Seats in a Hypothetical Three-State
House Delegation
Seat
Priority Values
House Size
State
Assignment
Multiplier (M)
Population (P)
(PxM)
4 CA 2
0.707106781
33,930,798
23,992,697.36
5 CA 3
0.40824829
33,930,798
13,852,190.28
6 NY 2
0.707106781
19,004,973
13,438,545.28
7 FL 2
0.707106781
16,028,890
11,334,136.81
8 CA 4
0.288675135
33,930,798
9,794,977.68
9 NY 3
0.40824829
19,004,973
7,758,747.74
10 CA 5
0.223606798
33,930,798
7,587,157.09
11 FL 3
0.40824829
16,028,890
6,543,766.94
12 CA 6
0.182574186
33,930,798
6,194,887.82
13 NY 4
0.288675135
19,004,973
5,486,263.14
14 CA 7
0.15430335
33,930,798
5,235,635.80
15 FL 4
0.288675135
16,028,890
4,627,141.98
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Seat
Priority Values
House Size
State
Assignment
Multiplier (M)
Population (P)
(PxM)
16 CA 8
0.133630621
33,930,798
4,534,193.61
17 NY 5
0.223606798
19,004,973
4,249,641.15
18 CA 9
0.11785113
33,930,798
3,998,782.89
19 FL 5
0.223606798
16,028,890
3,584,168.76
20 CA 10
0.105409255
33,930,798
3,576,620.15
21 NY 6
0.182574186
19,004,973
3,469,817.47
22 CA 11
0.095346259
33,930,798
3,235,174.65
23 CA 12
0.087038828
33,930,798
2,953,296.89
24 NY 7
0.15430335
19,004,973
2,932,531.00
25 FL 6
0.182574186
16,028,890
2,926,461.54
26 CA 13
0.080064077
33,930,798
2,716,638.02
27 NY 8
0.133630621
19,004,973
2,539,646.34
28 CA 14
0.074124932
33,930,798
2,515,118.08
29 FL 7
0.15430335
16,028,890
2,473,311.42
30 CA 15
0.069006556
33,930,798
2,341,447.51






Notes: The Constitution requires that each state have at least one seat. Consequently, the first three seats
assigned are not included in the table. Table prepared by CRS.
Next, the fifth seat’s allocation is determined. California’s claim to a third seat, based on the
computed priority value, is 13,852,190.28 (0.40824829 x 33,930,798), while, as above, New
York’s claim to its second seat is 13,438,545.28 (0.70710681 x 19,004,973) and Florida’s claim to
its second seat is 11,334,136.81 (0.70710681 x 16,028890). Again, California has a higher
priority value, and is allocated its third seat, the fifth seat in the hypothesized chamber.
Next the sixth seat’s allocation is determined in the same fashion. California’s claim to a fourth
seat, based on the computed priority value, is 9,794,977.68 (0.288675135 x 33,930,798), while,
as above, New York’s claim to its second seat is 13,438,545.28 (0.70710681 x 19,004,973) and
Florida’s claim to its second seat is 11,334,136.81 (0.70710681 x 16,028890). As New York’s
priority value is higher than either California’s or Florida’s, it is allocated its second seat, the sixth
seat in the hypothesized chamber.
Next, the seventh seat’s allocation is determined. Again, California’s claim to a fourth seat, based
on the computed priority value, is 9,794,977.68 (0.288675135 x 33,930,798), while, having
received its second seat, New York’s claim to its third seat is 7,758,747.738 (0.40824829 x
19,004,973) and Florida’s claim to its second seat is 11,334,136.81 (0.70710681 x 16,028890). As
Florida’s priority value is higher than either of the other states, Florida is, finally, allocated its
second seat, the seventh seat in the hypothesized chamber. This same process is continued until
all 30 seats in this hypothesized House are allocated to the three states.
From Table 2, then, we see that if the United States were made up of three states and the House
size were to be set at 30 Members, California would have 15 seats, New York would have eight,
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and Florida would have seven. Any other size House can be determined by picking points in the
priority list and observing what the maximum size state delegation size would be for each state.
A priority listing for all 50 states based on the 2000 Census is in the Appendix to this report. It
shows priority rankings for the assignment of seats in a House ranging in size from 51 to 500
seats.
Challenges to the Current Formula
The equal proportions rule of rounding at the geometric mean results in differing rounding points,
depending on which numbers are chosen. For example, the geometric mean between 1 and 2 is
1.4142, and the geometric mean between 49 and 50 is 49.49747. Table 3, below, shows the
“rounding points” for assignments to the House using the equal proportions method for a state
delegation size of up to 60. The rounding points are listed between each delegation size because
they are the thresholds that must be passed in order for a state to be entitled to another seat. The
table illustrates that, as the delegation size of a state increases, larger fractions are necessary to
entitle the state to additional seats.
The fact that higher rounding points are necessary for states to obtain additional seats has led to
charges that the equal proportions formula favors small states at the expense of large states. In
Fair Representation, a 1982 study of congressional apportionment, authors M.L. Balinski and
H.P. Young concluded that if “the intent is to eliminate any systematic advantage to either the
small or the large, then only one method, first proposed by Daniel Webster in 1832, will do.”15
This method, called the Webster method in Fair Representation, is also referred to as the major
fractions method (major fractions uses the concept of the adjustable divisor as does equal
proportions, but rounds at the arithmetic mean [.5] rather than the geometric mean.) Balinski and
Young’s conclusion in favor of major fractions, however, contradicts a report of the National
Academy of Sciences (NAS) prepared at the request of House Speaker Nicholas Longworth in
1929. The NAS concluded that “the method of equal proportions is preferred by the committee
because it satisfies ... [certain tests], and because it occupies mathematically a neutral position
with respect to emphasis on larger and smaller states.”16

15 Fair Representation, pp. 3-4. (An earlier major work in this field was written by Laurence F. Schmeckebier,
Congressional Apportionment (Washington: The Brookings Institution, 1941). Daniel Webster proposed this method to
overcome the large-state bias in Jefferson’s discarded fractions method. Webster’s method was used three times, in the
reapportionments following the 1840, 1910, and 1930 Censuses.
16 “Report of the National Academy of Sciences Committee on Apportionment” in The Decennial Population Census
and Congressional Apportionment, Appendix C, p. 21.
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Table 3. Rounding Points for Assigning Seats Using
the Equal Proportions Method of Apportionment
Size of
Round
Size of
Round
Size of
Round
Size of
Round
Delegation
Up At
Delegation
Up At
Delegation
Up At
Delegation
Up At
1 1.41421 16 16.49242 31 31.49603 46 46.49731
2 2.44949 17 17.49286 32 32.49615 47 47.49737
3 3.46410 18 18.49324 33 33.49627 48 48.49742
4 4.47214 19 19.49359 34 34.49638 49 49.49747
5 5.47723 20 20.49390 35 35.49648 50 50.49752
6 6.48074 21 21.49419 36 36.49658 51 51.49757
7 7.48331 22 22.49444 37 37.49667 52 52.49762
8 8.48528 23 23.49468 38 38.49675 53 53.49766
9 9.48683 24 24.49490 39 39.49684 54 54.49771
10 10.48809 25 25.49510 40 40.49691 55 55.49775
11 11.48913 26 26.49528 41 41.49699 56 56.49779
12 12.49000 27 27.49545 42 42.49706 57 57.49783
13 13.49074 28 28.49561 43 43.49713 58 58.49786
14 14.49138 29 29.49576 44 44.49719 59 59.49790
15 15.49193 30 30.49590 45 45.49725 60 60.49793
Notes: Any number between 645,684 and 645,930 divided into each state’s 2000 population will produce a
House size of 435 if rounded at these points, which are the geometric means of each pair of successive numbers.
Table prepared by CRS.
A bill that would have changed the apportionment method to another formula called the
“Hamilton-Vinton” method was introduced in 1981.17 The fundamental principle of the Hamilton-
Vinton method is that it ranks fractional remainders. In order to reapportion the House using
Hamilton-Vinton, each state’s population would be divided by the “ideal” sized congressional
district (in 2000, 281,424,177 divided by 435, for an “ideal” district population of 646,952). Any
state with fewer residents than the “ideal” sized district would receive a seat because the
Constitution requires each state to have at least one House seat. The remaining states in most
cases have a claim to a whole number and a fraction of a Representative. Each such state receives
the whole number of seats it is entitled to. The fractional remainders are rank-ordered from
highest to lowest until 435 seats are assigned. For the purpose of this analysis, we will
concentrate on the differences between the equal proportions and major fractions methods
because the Hamilton-Vinton method is subject to several mathematical anomalies.18

17 H.R. 1990, 97th Congress was introduced by Representative Floyd Fithian and was cosponsored by 10 other
Members of the Indiana delegation. Changing to the Hamilton-Vinton method would have kept Indiana from losing a
seat. Hearings were held, but no further action was taken on the measure. U.S. Congress, House Committee on Post
Office and Civil Service, Subcommittee on Census and Population, Census Activities and the Decennial Census,
hearing, 97th Cong., 1st sess., June 11, 1981, (Washington: GPO, 1981). Since that time no other bill has been introduced
to change the formula.
18 The Hamilton-Vinton method (used after the 1850-1900 censuses) is subject to the “Alabama paradox” and various
other population paradoxes. The Alabama paradox was so named in 1880 when it was discovered that Alabama would
have lost a seat in the House if the size of the House had been increased from 299 to 300. Another paradox, known as
(continued...)
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Equal Proportions or Major Fractions: An Analysis
Prior to the passage of the Apportionment Act of 1941 (2 U.S.C. 2(a)), the two contending
methods considered by Congress were the equal proportions method (Hill-Huntington) and the
method of major fractions (Webster). Each of the major competing methods—equal proportions
(currently used) and major fractions—can be supported mathematically. Choosing between them
is a policy decision, rather than a matter of conclusively proving that one approach is
mathematically better than the other. A major fractions apportionment results in a House in which
each citizen’s share of his or her Representative is as equal as possible on an absolute basis. In the
equal proportions apportionment now used, each citizen’s share of his or her Representative is as
equal as possible on a proportional basis. From a policy standpoint, a case can be made for either
method of computing the apportionment of seats by arguing that one measure of fairness is
preferable to the other.
The Case for Major Fractions
As noted above, a major fractions apportionment results in a House in which each person’s share
of his or her Representative is as equal as possible on an absolute basis. As an example, in 1990,19
the state of Massachusetts would have been assigned 11 seats under the major fractions method,
and the state of Oklahoma would have received 4 seats. Under this allocation, there would have
been 1.8245 Representatives per million for Massachusetts residents and 1.5835 Representatives
per million for Oklahoma residents. The absolute value20 of the difference between these two
numbers is 0.2410.
Under the equal proportion method of assigning seats in 1990, Massachusetts actually received 10
seats and Oklahoma 5. With 10 seats, Massachusetts received 1.6586 Representatives for each
million persons, and Oklahoma, with 5 seats, received 1.9002 Representatives per million
persons. The absolute value of the difference between these two numbers is 0.2415. As this
example shows, using the major fractions method produces a difference in the share of a
Representative between the states that is smaller, in an absolute sense, than is the difference
produced by the equal proportions method.
In addition, it can be argued that the major fractions minimization of absolute size differences
among districts more closely reflects the “one person, one vote” principle established by the
Supreme Court in its series of redistricting cases (Baker v. Carr, 369 U.S. 186 (1964) through
Karcher v. Daggett, 462 U.S.725 (1983).21

(...continued)
the population paradox, has been variously described, but in its modern form (with a fixed size House) it works in this
way: two states may gain population from one census to the next. State “A,” which is gaining population at a rate faster
than state “B,” may lose a seat to state “B.” There are other paradoxes of this type. Hamilton-Vinton is subject to them,
whereas equal proportions and major fractions are not.
19 The 2000 apportionment population is not used in this example due to the fact that there was no difference in the
apportionment of seats whether one used the method of major fractions or the method of equal proportions to allocate
seats to the states in 2000.
20 The absolute value of a number is its magnitude without regard to its sign. For example, the absolute value of -8 is 8.
The absolute value of the expression (4-2) is 2. The absolute value of the expression (2-4) is also 2.
21 Major fractions best conforms to the spirit of these decisions if the population discrepancy is measured on an
absolute basis, as the courts have done in the recent past. The Supreme Court has never applied its “one person, one
(continued...)
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Although the “one person, one vote” rules have not been applied by the courts to apportioning
seats among states, the method of major fractions can reduce the range between the smallest and
largest district sizes more than the method of equal proportions—one of the measures which the
courts have applied to within-state redistricting cases. Although this range would have not
changed in 2000 or 1990, if the method of major fractions had been used in 1980, the smallest
average district size in the country would have been 399,592 (one of Nevada’s two districts).
With the method of equal proportions it was 393,345 (one of Montana’s two districts). In both
cases the largest district was 690,178 (South Dakota’s single seat).22 Thus, in 1980, shifting from
equal proportions to major fractions as a method of apportionment would have improved the
296,833 difference between the largest and smallest districts by 6,247 persons. It can be argued,
because the equal proportions rounding points ascend as the number of seats increases, rather
than staying at .5, that small states may be favored in seat assignments at the expense of large
states. It is possible to demonstrate this by using simulation techniques.
The House has been reapportioned only 21 times since 1790. The equal proportions method has
been used in five apportionments, and the major fractions method in three. Eight apportionments
do not provide sufficient historical information to enable policy makers to generalize about the
impact of using differing methods. Computers, however, can enable reality to be simulated by
using random numbers to test many different hypothetical situations. These techniques (such as
the “Monte Carlo” simulation method) are a useful way to observe the behavior of systems when
experience does not provide sufficient information to generalize about them.
Apportioning the House can be viewed as a system with four main variables: (1) the size of the
House, (2) the population of the states,23 (3) the number of states,24 and (4) the method of
apportionment25. A 1984 exercise prepared for the Congressional Research Service (CRS)
involving 1,000 simulated apportionments examined the results when two of these variables were
changed—the method and the state populations. In order to further approximate reality, the state
populations used in the apportionments were based on the Census Bureau’s 1990 population
projections available at that time. Each method was tested by computing 1,000 apportionments
and tabulating the results by state. There was no discernible pattern by size of state in the results

(...continued)
vote” rule to apportioning seats of the House of Representatives among states (as opposed to redistricting within states).
Thus, no established rule of law is being violated. Arguably, no apportionment method can meet the “one person, one
vote” standard required by the Supreme Court for districts within states unless the size of the House is increased
significantly (thereby making districts less populous).
22 Nevada had two seats with a population of 799,184. Montana was assigned two seats with a population of 786,690.
South Dakota’s single seat was required by the Constitution (with a population of 690,178). The vast majority of the
districts based on the 1980 census (323 of them) fell within the range of 501,000 to 530,000).
23 For varying the definition of the population, see CRS Report RS22124, Potential House Apportionment Following
the 2010 Census Based on Census Bureau Population Projections, by Royce Crocker, and, Royce Crocker,
Apportioning Representatives Among States by Citizen Populations Instead of Total State Populations, CRS
Congressional Distribution Memorandum, May 8, 2007.
24 For information on the impact of adding states, see CRS Report RS22579, District of Columbia Representation:
Effect on House Apportionment, by Royce Crocker, and CRS Report R41113, Puerto Rican Statehood: Effects on
House Apportionment, by Royce Crocker.
25 See CRS Archived CRS Report RL31074, The House of Representatives Apportionment Formula: An Analysis of
Proposals for Change and Their Impact on States, by Royce Crocker.
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of the major fractions apportionment. The equal proportions exercise, however, showed that the
smaller states were persistently advantaged.26
Another way of evaluating the impact of a possible change in apportionment methods is to
determine the odds of an outcome being different than the one produced by the current method—
equal proportions. If equal proportions favors small states at the expense of large states, would
switching to major fractions, a method that appears not to be influenced by the size of a state,
increase the odds of the large states gaining additional representation? Based on the simulation
model prepared for CRS, this appears to be true. The odds of any of the 23 largest states gaining
an additional seat in any given apportionment range from a maximum of 13.4% of the time
(California) to a low of .2% of the time (Alabama). The odds of any of the 21 multi-districted
smaller states losing a seat range from a high of 17% (Montana, which then had two seats) to a
low of 0% (Colorado), if major fractions were used instead of equal proportions.
In the aggregate, switching from equal proportions to major fractions “could be expected to shift
zero seats about 37% of the time, to shift 1 seat about 49% of the time, 2 seats 12% of the time,
and 3 seats 2% of the time (and 4 or more seats almost never), and, these shifts will always be
from smaller states to larger states.”27
In summary, then, the method of major fractions minimizes the absolute differences in the share
of a representative between congressional districts across states. In addition, it appears that the
method of major fractions does not favor large or small states over the long term.
The Case for Equal Proportions
Support for the equal proportions formula primarily rests on the belief that minimizing the
proportional differences among districts is more important than minimizing the absolute
differences. Laurence Schmeckebier, a proponent of the equal proportions method, wrote in
Congressional Apportionment in 1941, that
Mathematicians generally agree that the significant feature of a difference is its relation to
the smaller number and not its absolute quantity. Thus the increase of 50 horsepower in the
output of two engines would not be of any significance if one engine already yielded 10,000
horsepower, but it would double the efficiency of a plant of only 50 horsepower. It has been
shown ... that the relative difference between two apportionments is always least if the
method of equal proportions is used. Moreover, the method of equal proportions is the only
one that uses relative differences, the methods of harmonic mean and major fraction being
based on absolute differences. In addition, the method of equal proportions gives the smallest
relative difference for both average population per district and individual share in a
representative. No other method takes account of both these factors. Therefore the method of
equal proportions gives the most equitable distribution of Representatives among the states.28

26 H.P. Young and M.L. Balinski, Evaluation of Apportionment Methods, Prepared under a contract for the
Congressional Research Service of the Library of Congress. (Contract No. CRS84-15), Sept. 30, 1984, available to
Members of Congress and congressional staff from the author of this report. Comparing equal proportions and major
fractions using the state populations from the 19 actual censuses taken since 1790, reveals that the small states would
have been favored 3.4% of the time if equal proportions had been used for all the apportionments. Major fractions
would have also favored small states, in these cases, but only .06 % of the time. See Fair Representation, p. 78.
27 Young and Balinski, Evaluation of Apportionment Methods, p. 13.
28 Schmeckebier, Congressional Apportionment, p. 60.
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An example using Massachusetts and Oklahoma 1990 populations, illustrates the argument for
proportional differences. The first step in making comparisons between the states is to standardize
the figures in some fashion. One way of doing this is to express each state’s representation in the
House as a number of Representatives per million residents.29 The equal proportions formula
assigned 10 seats to Massachusetts and 6 to Oklahoma in 1990. When 11 seats are assigned to
Massachusetts, and 5 are given to Oklahoma (using major fractions), Massachusetts has 1.824
Representatives per million persons and Oklahoma has 1.583 Representatives per million. The
absolute difference between these numbers is .241 and the proportional difference between the
two states’ Representatives per million is 15.22%. When 10 seats are assigned to Massachusetts
and 6 are assigned to Oklahoma (using equal proportions), Massachusetts has 1.659
Representatives per million and Oklahoma has 1.900 Representatives per million. The absolute
difference between these numbers is .243 and the proportional difference is 14.53%.
Major fractions minimizes absolute differences, so in 1990, if this if this method had been
required by law, Massachusetts and Oklahoma would have received 11 and 5 seats respectively
because the absolute difference (0.241 Representatives per million) is smaller at 11 and 5 than it
would be at 10 and 6 (0.243). Equal proportions minimizes differences on a proportional basis, so
it assigned 10 seats to Massachusetts and 6 to Oklahoma because the proportional difference
between a 10 and 6 allocation (14.53%) is smaller than would occur with an 11 and 5 assignment
(15.22%).
The proportional difference versus absolute difference argument could also be cast in terms of the
goal of “one person, one vote,” as noted above. The courts’ use of absolute difference measures in
state redistricting cases may not necessarily be appropriate when applied to the apportionment of
seats among states. The courts already recognize that the rules governing redistricting in state
legislatures differ from those in congressional districting. If the “one person, one vote” standard
were ever to be applied to apportionment of seats among states—a process that differs
significantly from redistricting within states—proportional difference measures might be
accepted as most appropriate.30
If the choice between methods were judged to be a tossup with regard to which mathematical
process is fairest, are there other representational goals that equal proportions meets which are,
perhaps, appropriate to consider? One such goal might be the desirability of avoiding large
districts, if possible. After the apportionment of 2000, five of the seven states with only one
Representative (Alaska, Delaware, Montana, North Dakota, South Dakota, Vermont, and

29 Representatives per million is computed by dividing the number of Representatives assigned to the state by the
state’s population (which gives the number of Representatives per person) and then multiplying the resulting dividend
by 1,000,000.
30 Montana argued in Federal court in 1991 and 1992 that the equal proportions formula violated the Constitution
because it “does not achieve the greatest possible equality in number of individuals per Representative” Department of
Commerce v. Montana 503 U.S. 442 (1992). Writing for a unanimous court, Justice Stevens however, noted that
absolute and relative differences in district sizes are identical when considering deviations in district populations within
states, but they are different when comparing district populations among states. Justice Stevens noted, however,
“although common sense” supports a test requiring a “good faith effort to achieve precise mathematical equality within
each State ... the constraints imposed by Article I, §2, itself make that goal illusory for the nation as a whole.” He
concluded “that Congress had ample power to enact the statutory procedure in 1941 and to apply the method of equal
proportions after the 1990 census.”
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Wyoming) have relatively large land areas. 31 The five Representatives of the larger states served
1.22% of the U.S. population, but also represented 27% of the U.S. total land area.
Arguably, an apportionment method that would potentially reduce the number of very large
districts would serve to increase representation in those states. Very large districts limit the
opportunities of constituents to see their Representatives, may require more district based offices,
and may require toll calls for telephone contact with the Representatives’ district offices.
Switching from equal proportions to major fractions may increase the number of states
represented by only one Member of Congress, although it is impossible to predict this outcome
with any certainty using Census Bureau projections for 2025.32
The table that follows contains the priority listing used in apportionment following the 2000
Census. Table A-1 shows where each state ranked in the priority of seat assignments. The priority
values listed beyond seat number 435 show which states would have gained additional
representations if the House size had been increased.

31 The total area of the U.S. is 3,795,951 square miles. The area and (rank) among all states in area for the seven single
district states in this scenario are as follows: Alaska−664,988 (1), Delaware−2,489 (49), Montana−147,039 (4), North
Dakota−70,698 (19), South Dakota−77,116 (17), Vermont−9,616 (45), Wyoming−97,812 (10), Source: U.S.
Department of Commerce, U.S. Census Bureau, Statistical Abstract of the United States, 2010, (Washington: GPO,
2010), Table 346: Land and Water Area of the States and Other Entities: 2008, p. 215.
32 U.S. Census Bureau, Projections of the Total Population of States: 1995-2025, Series A, http://www.census.gov/
population/projections/stpjpop.txt.
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Appendix. 2000 Priority List for Apportioning Seats
to the House of Representatives
Table A-1. 2000 Priority List for Apportioning Seats to the House of Representatives
Seat Sequence
State
Seat Number
Priority Value
51 California
2
23,992,697.36
52 Texas
2
14,781,355.91
53 California
3
13,852,190.28
54 New
York
2
13,438,545.28
55 Florida
2
11,334,136.81
56 California
4
9,794,977.68
57 Illinois 2
8,795,730.95
58 Pennsylvania
2
8,697,887.17
59 Texas
3
8,534,019.81
60 Ohio
2
8,043,014.37
61 New
York
3
7,758,747.74
62 California
5
7,587,157.09
63 Michigan
2
7,039,834.20
64 Florida
3
6,543,766.94
65 California
6
6,194,887.82
66 Texas
4
6,034,463.28
67 New
Jersey
2
5,956,917.84
68 Georgia
2
5,803,207.68
69 North
Carolina
2
5,704,706.29
70 New
York
4
5,486,263.14
71 California
7
5,235,635.80
72 Illinois 3
5,078,217.63
73 Pennsylvania
3
5,021,727.50
74 Virginia
2
5,020,954.54
75 Texas
5
4,674,275.16
76 Ohio
3
4,643,636.51
77 Florida
4
4,627,141.98
78 California
8
4,534,193.61
79 Massachusetts
2
4,494,065.23
80 Indiana
2
4,306,833.25
81 New
York
5
4,249,641.15
82 Washington
2
4,178,070.52
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Seat Sequence
State
Seat Number
Priority Value
83 Michigan
3
4,064,450.17
84 Tennessee
2
4,030,534.82
85 California
9
3,998,782.89
86 Missouri
2
3,964,224.46
87 Texas
6
3,816,529.69
88 Wisconsin
2
3,798,019.01
89 Maryland
2
3,753,242.18
90 Arizona
2
3,635,011.81
91 Illinois 4
3,590,842.12
92 Florida
5
3,584,168.76
93 California
10
3,576,620.15
94 Pennsylvania
4
3,550,897.57
95 Minnesota
2
3,482,974.66
96 New
York
6
3,469,817.47
97 New
Jersey
3
3,439,228.12
98 Georgia
3
3,350,483.51
99 North
Carolina
3
3,293,613.71
100 Ohio
4
3,283,546.87
101 California
11
3,235,174.65
102 Texas
7
3,225,556.30
103 Louisiana
2
3,168,030.01
104 Alabama
2
3,154,495.27
105 Colorado
2
3,048,961.00
106 California
12
2,953,296.89
107 New
York
7
2,932,531.00
108 Florida
6
2,926,461.54
109 Virginia
3
2,898,849.45
110 Michigan
4
2,874,000.28
111 Kentucky
2
2,863,380.12
112 South
Carolina
2
2,846,147.93
113 Texas
8
2,793,413.70
114 Illinois 5
2,781,454.35
115 Pennsylvania
5
2,750,513.43
116 California
13
2,716,638.02
117 Massachusetts
3
2,594,649.77
118 Ohio
5
2,543,424.47
119 New
York
8
2,539,646.34
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Seat Sequence
State
Seat Number
Priority Value
120 California
14
2,515,118.08
121 Indiana
3
2,486,551.34
122 Florida
7
2,473,311.42
123 Texas
9
2,463,559.32
124 Oklahoma
2
2,445,754.37
125 New
Jersey
4
2,431,901.52
126 Oregon
2
2,424,346.00
127 Washington
3
2,412,210.14
128 Connecticut
2
2,410,905.32
129 Georgia
4
2,369,149.61
130 California
15
2,341,447.51
131 North
Carolina
4
2,328,936.59
132 Tennessee
3
2,327,030.36
133 Missouri
3
2,288,746.06
134 Illinois 6
2,271,047.97
135 Pennsylvania
6
2,245,784.81
136 New
York
9
2,239,757.55
137 Michigan
5
2,226,191.04
138 Texas
10
2,203,474.44
139 Wisconsin
3
2,192,787.30
140 California
16
2,190,223.59
141 Maryland
3
2,166,935.39
142 Florida
8
2,141,950.52
143 Arizona
3
2,098,675.05
144 Ohio
6
2,076,697.38
145 Iowa
2
2,073,182.64
146 California
17
2,057,356.83
147 Virginia
4
2,049,796.11
148 Mississippi
2
2,017,324.03
149 Minnesota
3
2,010,896.36
150 New
York
10
2,003,300.05
151 Texas
11
1,993,117.62
152 California
18
1,939,694.62
153 Illinois 7
1,919,385.85
154 Kansas
2
1,904,821.22
155 Pennsylvania
7
1,898,034.59
156 Arkansas
2
1,894,857.38
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Seat Sequence
State
Seat Number
Priority Value
157 Florida
9
1,889,022.80
158 New
Jersey
5
1,883,742.82
159 Georgia
5
1,835,135.40
160 California
19
1,834,767.42
161 Massachusetts
4
1,834,694.45
162 Louisiana
3
1,829,062.98
163 Alabama
3
1,821,248.70
164 Texas
12
1,819,459.14
165 Michigan
6
1,817,677.37
166 New
York
11
1,812,053.08
167 North
Carolina
5
1,803,986.52
168 Colorado
3
1,760,318.46
169 Indiana
4
1,758,257.31
170 Ohio
7
1,755,129.63
171 California
20
1,740,613.21
172 Washington
4
1,705,690.15
173 Florida
10
1,689,593.36
174 Texas
13
1,673,658.98
175 Illinois 8
1,662,236.91
176 California
21
1,655,653.41
177 New
York
12
1,654,170.58
178 Kentucky
3
1,653,173.28
179 Tennessee
4
1,645,458.95
180 Pennsylvania
8
1,643,746.17
181 South
Carolina
3
1,643,224.27
182 Missouri
4
1,618,387.86
183 Virginia
5
1,587,765.24
184 Utah
2
1,581,595.64
185 California
22
1,578,603.59
186 Wisconsin
4
1,550,534.77
187 Texas
14
1,549,507.13
188 New
Jersey
6
1,538,069.57
189 Michigan
7
1,536,217.77
190 Maryland
4
1,532,254.71
191 Florida
11
1,528,294.70
192 New
York
13
1,521,615.62
193 Ohio
8
1,519,986.84
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Seat Sequence
State
Seat Number
Priority Value
194 California
23
1,508,407.96
195 Georgia
6
1,498,381.78
196 Arizona
4
1,483,987.36
197 North
Carolina
6
1,472,948.83
198 Illinois 9
1,465,955.16
199 Pennsylvania
9
1,449,647.86
200 California
24
1,444,190.67
201 Texas
15
1,442,512.63
202 Minnesota
4
1,421,918.45
203 Massachusetts
5
1,421,148.21
204 Nevada
2
1,415,650.40
205 Oklahoma
3
1,412,056.94
206 New
York
14
1,408,742.32
207 Oregon
3
1,399,696.82
208 Florida
12
1,395,135.80
209 Connecticut
3
1,391,936.84
210 California
25
1,385,219.03
211 Indiana
5
1,361,940.26
212 Texas
16
1,349,347.01
213 Ohio
9
1,340,502.39
214 California
26
1,330,875.39
215 Michigan
8
1,330,403.61
216 Washington
5
1,321,221.91
217 New
York
15
1,311,467.73
218 Illinois 10
1,311,190.15
219 New
Jersey
7
1,299,906.04
220 Pennsylvania
10
1,296,604.46
221 Virginia
6
1,296,404.89
222 Louisiana
4
1,293,342.83
223 New
Mexico
2
1,289,636.20
224 Alabama
4
1,287,817.30
225 Florida
13
1,283,338.28
226 West
Virginia
2
1,282,039.04
227 California
27
1,280,635.44
228 Tennessee
5
1,274,567.02
229 Texas
17
1,267,490.81
230 Georgia
7
1,266,363.74
Congressional Research Service
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The U.S. House of Representatives Apportionment Formula in Theory and Practice

Seat Sequence
State
Seat Number
Priority Value
231 Missouri
5
1,253,597.85
232 North
Carolina
7
1,244,868.97
233 Colorado
4
1,244,733.12
234 California
28
1,234,051.19
235 New
York
16
1,226,765.73
236 Nebraska
2
1,212,949.05
237 Wisconsin
5
1,201,039.07
238 Ohio
10
1,198,981.79
239 Iowa
3
1,196,952.55
240 Texas
18
1,195,001.80
241 California
29
1,190,737.58
242 Florida
14
1,188,140.38
243 Maryland
5
1,186,879.39
244 Illinois 11
1,186,016.12
245 Michigan
9
1,173,305.70
246 Pennsylvania
11
1,172,822.87
247 Kentucky
4
1,168,970.04
248 Mississippi
3
1,164,702.57
249 South
Carolina
4
1,161,935.03
250 Massachusetts
6
1,160,362.65
251 New
York
17
1,152,345.75
252 California
30
1,150,361.79
253 Arizona
5
1,149,491.66
254 Texas
19
1,130,358.54
255 New
Jersey
8
1,125,751.66
256 California
31
1,112,634.70
257 Indiana
6
1,112,019.56
258 Florida
15
1,106,098.49
259 Minnesota
5
1,101,413.30
260 Kansas
3
1,099,749.04
261 Georgia
8
1,096,703.17
262 Virginia
7
1,095,662.11
263 Arkansas
3
1,093,996.42
264 New
York
18
1,086,441.99
265 Ohio
11
1,084,519.84
266 Illinois 12
1,082,679.64
267 Washington
6
1,078,773.17
Congressional Research Service
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The U.S. House of Representatives Apportionment Formula in Theory and Practice

Seat Sequence
State
Seat Number
Priority Value
268 North
Carolina
8
1,078,088.15
269 California
32
1,077,303.91
270 Texas
20
1,072,352.27
271 Pennsylvania
12
1,070,635.90
272 Michigan
10
1,049,436.52
273 California
33
1,044,148.13
274 Tennessee
6
1,040,679.61
275 Florida
16
1,034,660.40
276 New
York
19
1,027,671.24
277 Missouri
6
1,023,558.36
278 Texas
21
1,020,010.46
279 California
34
1,012,972.48
280 Louisiana
5
1,001,819.05
281 Oklahoma
4
998,475.04
282 Alabama
5
997,538.99
283 Illinois 13
995,920.42
284 New
Jersey
9
992,819.64
285 Ohio
12
990,026.63
286 Oregon
4
989,735.11
287 Pennsylvania
13
984,841.79
288 Connecticut
4
984,247.98
289 California
35
983,604.69
290 Massachusetts
7
980,685.43
291 Wisconsin
6
980,644.29
292 New
York
20
974,934.54
293 Texas
22
972,541.82
294 Florida
17
971,894.21
295 Maryland
6
969,082.96
296 Georgia
9
967,201.28
297 Colorado
5
964,166.13
298 California
36
955,891.94
299 North
Carolina
9
950,784.38
300 Michigan
11
949,251.05
301 Virginia
8
948,871.22
302 Indiana
7
939,828.07
303 Arizona
6
938,556.01
304 California
37
929,698.14
Congressional Research Service
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The U.S. House of Representatives Apportionment Formula in Theory and Practice

Seat Sequence
State
Seat Number
Priority Value
305 Texas
23
929,295.88
306 New
York
21
927,347.73
307 Illinois 14
922,043.14
308 Idaho
2
917,311.24
309 Florida
18
916,310.65
310 Utah
3
913,134.67
311 Pennsylvania
14
911,786.32
312 Washington
7
911,729.74
313 Ohio
13
910,692.05
314 Kentucky
5
905,480.30
315 California
38
904,901.72
316 Maine
2
903,492.25
317 South
Carolina
5
900,031.00
318 Minnesota
6
899,300.19
319 Texas
24
889,733.07
320 New
Jersey
10
888,004.88
321 New
York
22
884,191.36
322 California
39
881,393.76
323 Tennessee
7
879,534.80
324 New
Hampshire
2
875,691.64
325 Florida
19
866,743.10
326 Michigan
12
866,543.69
327 Georgia
10
865,091.12
328 Missouri
7
865,064.70
329 Hawai
2
860,295.81
330 California
40
859,076.37
331 Illinois 15
858,375.45
332 Texas
25
853,401.98
333 North
Carolina
10
850,407.40
334 Massachusetts
8
849,298.50
335 Pennsylvania
15
848,826.87
336 Iowa
4
846,373.27
337 New
York
23
844,874.10
338 Ohio
14
843,137.00
339 California
41
837,861.34
340 Virginia
9
836,825.76
341 Wisconsin
7
828,795.70
Congressional Research Service
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The U.S. House of Representatives Apportionment Formula in Theory and Practice

Seat Sequence
State
Seat Number
Priority Value
342 Mississippi
4
823,569.09
343 Florida
20
822,264.71
344 Texas
26
819,922.10
345 Maryland
7
819,024.59
346 Louisiana
6
817,981.83
347 California
42
817,668.94
348 Nevada
3
817,326.14
349 Alabama
6
814,487.18
350 Indiana
8
813,914.98
351 New
York
24
808,905.37
352 New
Jersey
11
803,230.64
353 Illinois 16
802,936.71
354 California
43
798,426.97
355 Michigan
13
797,104.26
356 Pennsylvania
16
794,004.83
357 Arizona
7
793,224.61
358 Washington
8
789,581.11
359 Texas
27
788,970.41
360 Colorado
6
787,238.35
361 Ohio
15
784,917.83
362 Georgia
11
782,504.36
363 Florida
21
782,129.75
364 California
44
780,069.88
365 Kansas
4
777,640.01
366 New
York
25
775,874.77
367 Arkansas
4
773,572.28
368 Oklahoma
5
773,415.44
369 North
Carolina
11
769,222.44
370 Oregon
5
766,645.52
371 California
45
762,537.98
372 Connecticut
5
762,395.20
373 Tennessee
8
761,699.48
374 Texas
28
760,270.91
375 Minnesota
7
760,047.38
376 Illinois 17
754,227.71
377 Missouri
8
749,168.01
378 Massachusetts
9
749,010.87
Congressional Research Service
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The U.S. House of Representatives Apportionment Formula in Theory and Practice

Seat Sequence
State
Seat Number
Priority Value
379 Virginia
10
748,479.71
380 Pennsylvania
17
745,837.67
381 California
46
745,776.85
382 Florida
22
745,731.45
383 New
York
26
745,436.37
384 New
Mexico
3
744,571.81
385 Rhode
Island
2
742,223.12
386 West
Virginia
3
740,185.59
387 Kentucky
6
739,321.57
388 Michigan
14
737,975.14
389 South
Carolina
6
734,872.24
390 Ohio
16
734,223.40
391 Texas
29
733,586.38
392 New
Jersey
12
733,245.90
393 California
47
729,736.77
394 Indiana
9
717,805.54
395 Wisconsin
8
717,758.13
396 New
York
27
717,296.48
397 California
48
714,372.17
398 Georgia
12
714,325.49
399 Florida
23
712,571.08
400 Illinois 18
711,092.70
401 Maryland
8
709,296.10
402 Texas
30
708,711.77
403 Pennsylvania
18
703,182.50
404 North
Carolina
12
702,200.80
405 Nebraska
3
700,296.46
406 California
49
699,641.26
407 Washington
9
696,345.09
408 Louisiana
7
691,320.82
409 New
York
28
691,204.19
410 Ohio
17
689,682.79
411 Alabama
7
688,367.30
412 Michigan
15
687,017.47
413 Arizona
8
686,952.66
414 California
50
685,505.64
415 Texas
31
685,468.97
Congressional Research Service
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The U.S. House of Representatives Apportionment Formula in Theory and Practice

Seat Sequence
State
Seat Number
Priority Value
416 Florida
24
682,234.86
417 Virginia
11
677,025.37
418 New
Jersey
13
674,488.13
419 Illinois 19
672,626.36
420 California
51
671,929.93
421 Tennessee
9
671,755.80
422 Massachusetts
10
669,935.69
423 New
York
29
666,943.80
424 Colorado
7
665,337.84
425 Pennsylvania
19
665,144.06
426 Texas
32
663,702.47
427 Missouri
9
660,704.08
428 California
52
658,881.50
429 Minnesota
8
658,220.34
430 Georgia
13
657,083.88
431 Iowa
5
655,597.91
432 Florida
25
654,376.69
433 Ohio
18
650,239.17
434 California
53
646,330.23
435 North
Carolina
13
645,930.79
Last seat assigned by current law
436 Utah
4
645,683.72
437 New
York
30
644,328.93
438 Texas
33
643,275.95
439 Michigan
16
642,646.00
440 Indiana
10
642,024.80
441 Montana
2
640,155.08
442 Illinois 20
638,109.39
443 Mississippi
5
637,933.87
444 California
54
634,248.22
445 Wisconsin
9
633,003.17
446 Oklahoma
6
631,491.06
447 Pennsylvania
20
631,011.06
448 Florida
26
628,704.79
449 Oregon
6
625,963.45
450 Maryland
9
625,540.36
451 Kentucky
7
624,840.77
Congressional Research Service
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The U.S. House of Representatives Apportionment Formula in Theory and Practice

Seat Sequence
State
Seat Number
Priority Value
452 New
Jersey
14
624,454.66
453 Texas
34
624,069.34
454 New
York
31
623,197.62
455 Washington
10
622,829.98
456 California
55
622,609.65
457 Connecticut
6
622,493.08
458 South
Carolina
7
621,080.40
459 Virginia
12
618,036.78
460 Ohio
19
615,064.68
461 California
56
611,390.54
462 Georgia
14
608,341.46
463 Illinois 21
606,963.10
464 Massachusetts
11
605,979.63
465 Texas
35
605,976.51
466 Arizona
9
605,835.30
467 Florida
27
604,971.47
468 Michigan
17
603,660.80
469 New
York
32
603,408.50
470 Kansas
5
602,357.36
471 Tennessee
10
600,836.66
472 California
57
600,568.61
473 Pennsylvania
21
600,211.24
474 Arkansas
5
599,206.51
475 Louisiana
8
598,701.40
476 North
Carolina
14
598,015.71
477 Alabama
8
596,143.57
478 Missouri
10
590,951.69
479 California
58
590,123.15
480 Texas
36
588,903.32
481 New
York
33
584,837.62
482 Ohio
20
583,501.59
483 Florida
28
582,965.09
484 New
Jersey
15
581,335.66
485 Indiana
11
580,733.28
486 Minnesota
9
580,495.78
487 California
59
580,034.83
488 Illinois 22
578,716.61
Congressional Research Service
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The U.S. House of Representatives Apportionment Formula in Theory and Practice

Seat Sequence
State
Seat Number
Priority Value
489 Nevada
4
577,936.86
490 Colorado
8
576,199.47
491 Texas
37
572,765.91
492 Pennsylvania
22
572,278.96
493 California
60
570,285.65
494 Michigan
18
569,136.86
495 Virginia
13
568,511.15
496 New
York
34
567,375.83
497 Georgia
15
566,335.08
498 Wisconsin
10
566,175.25
499 Washington
11
563,370.91
500 Florida
29
562,503.77




Notes: Prepared by CRS.

Author Contact Information

Royce Crocker

Specialist in American National Government
rcrocker@crs.loc.gov, 7-7871


Congressional Research Service
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